I know for a fact that this game has a lot of item randomization, and items are what you need to win the game, in which I really haven't figured out how that works yet, but I know that the shops and looking in drawers are two good examples of this.
Now, looking at an FAQ on GameFAQs, I came up with an item list. I left the original FAQ description intact, but I'm adding in my own thoughts since I've gotten to the end of the game myself.
Wallet: Used to purchase items at the store.
Useful?: I believe that it's absolutely mandatory to have, plus it's easy to get. You need it to at least buy the detonator at the shop so you can get into Toontown.
Fish Bones: Used to lure the cats away from an item.
Useful?: Possibly, if you can manage the luck manipulation so that you get a useful item rather than garbage.
Meat: Used to lure dogs away from an item.
Useful?: See "Fish Bones"
Rattle: Used to lure snakes away from an item.
Useful?: I believe it's mandatory, since you need it in the cave to get a piece of the will. It's been a while since I've played this.
Baseball: Used to distract the weasel at the factory.
Useful: Mandatory. You're not goin' anywhere in the factory without it!
Bomb: Used as a weapon.
Boxing Glove: Used as a weapon.
Brick: Used as a weapon.
Custard Pie: Used as a weapon.
Cigar: Used as a weapon.
Six-shooter: Used as a weapon.
Useful?: Maybe. I don't know how quick they fight off Psycho (Who randomly appears outside buildings), and they'd be good with Judge Doom.
In fact, it might be possible to fix it so that you never run into Psycho and have to fight him, so they'd only be good on Judge Doom.
Crowbar: Used as a weapon.
Useful?: I believe this has a secondary use of opening doors with locks on them. In a speed run, I don't know how necessary opening locked doors would be.
Rose: Used to get tips from Jessica Rabbit.
Useful: Not particularly. The only "tip" I remember getting from her is her telling you to find her phone number and give her a call. You can find her phone number, but the item doesn't work anymore, I believe. She used to give some really good beginner hints, but why they cut that out of the game, I have no idea. I'm not sure if I'm wrong about this, but if it works, make a little movie with you calling Jessica and show me.
Heart: Used to get tips from lady in hotel room.
Useful?: Nope, not in a speed run.
Detonator: Used to explode the dynamite.
Useful?: Absolutely necessary.
Dynamite: Blows up wall to Toontown
Useful?: You know you need it. It's lying around in random places if I remember right. If not, it's definitely found in the shop.
Spring Shoes: Used to jump holes in the caves.
Useful?: Mandatory to get at least one piece of the will in the caves.
The Will: Allows entrance to factory.
Useful?: Duh.
Password: Used to get into Ink & Paint Club.
Useful?: Not in a speed run, no. Going in to talk to Jessica isn't needed at all in the game, speed run or not.
Whistle Used to hail Benny The Cab.
Useful?: In a speed run, absolutely. Benny traverses the roads lightning quick. However, he won't go off the roads, so there's still going to be quite a bit of walking. However, Benny makes the time spent moving around the overworld less.
Flashlight: Used in caves to give light.
Useful?: I don't know if you can just skip using this or not in the caves, I never tried it. It's a possibility, seeing as how game designers didn't usually think that far ahead back in those days, but I could be wrong.
Portable Hole: Makes a quick end to a Psycho fight
Useful?: If the Psycho fights absolutely have to happen, then yes, this would be a must-have item for just getting rid of him quickly. (Thanks, Mr. Collins. =D)
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So anyway, the mandatory items in this game are:
-The Whistle
-The Will
-The Dynamite
-The Detonator
-The Rattle
-The Spring Shoes
-The Baseball
-The Wallet
The possibly mandatory items in this game (In order of likliness) are:
-The Flashlight
-The Meat
-The Fish Bones
-The Weapon Items
-The Crowbar
-The Portable Hole
With the meat and fish bones, there may be a way to dodge the dog or cat that guards the item with some possible optimization, and maybe a trick or glitch or two, perhaps involving the Spring Shoes. That would render the meat and fish bones useless if you can get the item and get out of there alive. When you're close enough to a "guarding" animal, when you get close, they'll make an insta-dash towards you, and one hit kills you.
Speaking of deaths, they're definitely not anything that is worth using for shortcuts. Deaths should be avoided at all costs, since all it does it make you lose time, and you can leave the screen anywhere via the bottom.
For Judge Doom himself, I've never actually beaten him yet. I used a Game Genie to play against him, so that's why the final battle may have been glitchy, but someone give me a different experience if this isn't what happened.
I basically punched and pounded on the fellow until he finally fell down. When he was down, I ran over to the dip shooter and grabbed it. Now, as soon as I set foot on the ground, he would basically make an insta-charge at me so fast, and the dip shooter never seemed to hit him, so I'd always die.
However, one time I actually dodged him by going up the ladder and he just missed me. He ran off the other side all kamikaze-like and he actually disappeared for about a five or ten minute span...then he came back around from behind and hit me and killed me. I got mad at the game and didn't play it for a while. XD
So anyway, that's some of the advice I can offer to anyone doing a speed run of the game, since I can't do it myself. I'll try to get my hands on some more useful info later on if there's further interest in the game.