Banjo-Kazooie

First off, the quickest route to get to the Furnace Fun Quiz Show from the beginning of Grunty's Lair:

Go left at the start up the steep slope, past the bee hive, go through the door, and make an immediate left (banjo's left, your right since the screen view is looking backwards when you enter the room), though the next door to jump into the purple cauldron. Once you exit the cauldron, follow the wall around to the right (again Banjo's right, not yours) and go up a steep ramp. Enter the door at the top of the ramp, and you'll find yourself swimming. Quickly (or not) make your way to the surface, and jump up the ledges nearly straight across the room from you to another door. In this tunnel either destroy the roots with egg bombs, or just try to avoid them. Once you make it to the open field, turn right (this time Banjo's right is the same as yours if you didn't mess the camera up), and jump up the leaf ledges, and enter the door at the top. Finally step on the Tooty face, and you'll be transported to the start of the puzzle.

To get the answers for your in-game specific questions about Gruntilda for the trivia questions (the game board space is a picture of Gruntilda), you'll have to locate the 10 Brunhildas in your world. Her locations and possible repsonses are listed below. When you locate her, the vibrating text is the correct response of the three possible in your game.

*Brentilda 1: The 1st Brentilda is located in room with the Treasure Trove Cove and Clanker's Cavern jigsaws, just past the 50 Note Door.
1. Grunty brushes her rotten teeth with salted slug/moldy cheese/tuna ice cream flavored tooth paste!
2. She also washes her hair with engine oil/rancid milk/baked beans. Yuk!
3. And she gets her clothes from the trash can/The Witches Warehouse/Saggy Maggy's Boutique!

*Brentilda 2: The 2nd Brentilda is located in the Click Clock Wood puzzle room.
1. Ugly Grunty's nickname was Jelly Belly/Cauldron Butt/Hog Breath at witch school!
2. I also know that freshly burst boils/sweaty gorilla feet/putrid parrot puke is her favorite smell!
3. And the old hag's favorite color is ghastly grey/dung brown/gruesome green!

* Brentilda 3: The 3rd Brentilda is located in Clanker's Cavern entrance on top of the large horizontal pipe.
1. Grunty wears a flea circus/a reinforced girdle/massive bloomers under that repulsive dress of hers!
2. She's also got this nasty pet dog whose name is Ripper/Legchomper/Bignacka!
3. My sister sings in her own band, Grunty and the Cauldron Crew/Grunty and the Broomstick Boys/Grunty and the Monster Mob. They're awful!

*Brentilda 4: The 4th Brentilda is located behind the Huge Grunty Statue.
1. My lazy sister often sleeps in a pig sty/on a pile of treasure/in a dumpster, the dirty hag!
2. The only thing she's ever won was the biggest butt/dirtiest undies/sweatiest socks competition at witch school!
3. She often boasts of appearing on the cover of Fat Hag Monthly, posing in her long-johns/sitting on her broomstick/polishing her crystal ball!

*Brentilda 5: The 5th Brentilda is located inside one of the logs in the Bubble Gloop Swamp Entrance room.
1. My fat old sister's favorite sport is broomstick racing/belly barging/loogie flicking!
2. Although she's dim, she attended Fat Hag High/St. Dungball's School/Cauldron College!
3. You won't believe that Gruntilda's party trick is blowing balloons up with her butt/performing a scary striptease/eating a bucket of beans!

*Brentilda 6: The 6th Brentilda is located behind the entrance to Gobi's Valley.
1. Grunty's best friend at witch school was the awful Fatty Hatty/Sweaty Betty/Saggy Maggy!
2. When relaxing, she usually reads Fat Hag Monthly/Big Butts and Guts/Warty Girls Weekly magazine!
3. While sipping a glass of her favorite camel spit milkshake/cold worm juice/smoothie elephant sweat!

*Brentilda 7: The 7th Brentilda is located on top of the Grunty Face Statue's hat.
1. The disgusting Gruntilda has spider pancakes/rat bagels/tadpoles on toast for breakfast!
2. Then she usually has slug stew/maggot pie/dog dung burgers for dinner. Yuk!
3. Wartbags then finishes with eyeball ice cream/cockroaches and cream/rat sorbet for dessert. How horrid!

*Brentilda 8: The 8th Brentilda is located on the ledge that the narrow path in the Lava Room leads to.
1. Revolting Gruntilda's bedroom has smelly socks/rotting fish/dirty undies hanging from the ceiling!
2. She also has an eyeball flower/a loogie bush/a verruca plant growing in a pot beside her bed! Filthy old bag!
3. And you'd be sick if you saw her enormous spotty purple/sweaty yellow/streaky brown undies!

*Brentilda 9: The 9th Brentilda is located behind one of the gravestones to the right of the Mad Monster Mansion entrance.
1. Did you know Wartbags keeps a dragon's foot/loogie filled hanky/shrunken head in her pocket for luck?
2. I've also seen my sister cuddling a huge sweaty baboon/her dirty undies/a scary teddy bear in bed at night!
3. She's really proud of her broomstick. It's a top of the range Super Gutlifter Pro/Lardmaster 2000/Rough Rider Deluxe!

*Brentilda 10: The 10th Brentilda is located behind one of the tall logs in the Click Clock Wood entrance room.
1. Gruesome Gruntilda's favorite pastime is flying radio controlled bats/bursting boils/collecting dragon teeth!
2. This poor guy called Dirty Berty/Undead Ed/Greasy Grant, was her first and only boyfriend!
3. When she was younger, Grunty used to have a baby dragon/a greasy warthog/a mad vulture as a pet.

I found this at http://banjokazooie.wikia.com/wiki/Facts_about_Gruntilda# and is not my original work.

Regarding the miscellaneous questions (the ones with Banjo-Kazooie on the space), I used this guide:
http://banjokazooie.wikia.com/wiki/List_of_Questions_From_Gr...'s_Furnace_Fun

Since you only have 10 seconds to answer the question, and memorizing all these answers is pretty impossible, what I did was copy the answers into a text document (put in my answers for the Grunty trivia that you got from Bruntillda), and then would pull up a search window in the appropriate document (typically CTRL+F) and when she asked the question I quickly typed the most unique word from the question in the search box to quickly locate the line of importance. Doing this I was pretty successful. Only getting it wrong when she asked what was in the pot next to her bed, since the words were too short and I hit three incorrect questions prior to get the answer I needed.

I am still looking for a good reference for the pictures that are used for the eyeball spaces, and as far as the music note spaces, you only need to know the names of the characters, since the question is "which character's voice is this?", and the possible responses are typically variations of the character's name (e.g. for Clanker the Whale, the wrong answers are Clunker or Planker).

A final clue for doing this puzzle would be to pick up the joker squares. Most guides tell you to save your jokers for the skull spaces since it means instant death if you get them wrong, but I sucked at the mini games (the clock spaces) in this puzzle as well, so I set off to collect as many jokers as I could so I could skip all the skull and mini game spaces I came to.

The path I took through the maze is as follows:
1. Go straight, answer 2 questions
2. turn left answer two questions, and get a joker space
3. backtrack to the right and do the gunty question
4. go up 3 spaces and skip the timer
5. go right all the way and then down two spaces to another joker
6. backtrack back to the timer space that was already skipped, and continue left
7. skip the next stop watch space
8. continue left and up, skip the skull space
9. go left and then down to the next joker space
10. go up and then right, skipping the stop watch space
11. continue all the way up and then left to skip the skull space and continue to the joker
12. backtrack down to the two grunty questions that are next to each other and go right, skip the stop watch
13 continue up right up, you should have two jokers left to skip the remaining skulls, and the puzzle is done.

Bare in mind though, to skip all the stopwatch and skull spaces that I hit, I got all the joker questions correct on the first try. I don't know if those questions were actually easier, or if I just got lucky on the questions that I was asked. Most of them came from the same pool as used for the Banjo-Kazooie spaces, but I did get one that was for a character's voice. If you mess up one of the joker questions, there is a joker space above the first one that I directed you to, but there's no way to get to it without hitting an additional skull or stop watch space, so the net result would be that you'd only be short 1 joker instead of 2 providing you get that question correct.

Make sure you have a read of what Gruntilda's fairy sister tells you on your journey through the game. She can be found around the lair waving her magic wand, you can hear her distinct music when you pass by her. She can also replenish your health so thats an added bonus.

Each individual place she exists in she can provide you with 3 pieces of information. Have a vague recollection at least of what she said by the time you reach the quiz show.

You'll also need to remember parts of levels, soundtracks of levels, character's names, place's names and a few other things, so stay awake to avoid having to repeat the quiz several times over.

Keep your joker cards for the death squares symbolised by the skull on the quiz board.

Cant hack the boss fights?
Cant stand be bothered to memorise all of the crazy random questions about gruntys toothpaste choice?
Then this is the solution for you.

The game has a nice glitch which will allow you to skip pretty much all of this section , its a bit fiddly to pull off but pretty simple once you get the knack.

First of all enter the quiz area and answer the first two questions then take a left and answer the next three questions in a row.
This should leave you with a skull death square followed by a 1 up in front of you

Now the tricky part.
Walk slowly from the edge of the joker square towards the death square , there are a few pixels in between these two squares which allow banjo to use special moves.
By walking into this area and activating Talon Trot while walking, you can slide into the skull tile while the Talon Trot animation is being done. As you slide into the square with talon trot activating press a to bring up the question.
Providing the question isnt a picture question the next part will work.

Now get the question wrong. Ordinarily this would then fling you into the lava to die but here you simply bounce up in the air a little bit.
Now the cool bit. The game thinks you are dead and you can run around the board without any invisible walls preventing you from moving forward. This allows you to skip the rest of the questions and boss fights!
Now simply walk to the last square. Get the question correct and you get your achievement.

EDIT: As suggested by MadMikeForever, you can use the joker card you may have picked up before pulling off the glitch to skip the last question and still get the Achievement.