Reddit mentions of Carb Carburetor Jet Cleaner, Torch Tip Cleaner Tools for ATV Welder Motorcycle - 13 Wire Set, 2 PCS

Not an expert, but

I'd focus on the drive train issues first, get all those tires working, might be a lost cause before going any deeper, but others will have to help with that, I take that kind of stuff to my mechanic.

Brake pedal could be that it's low on brake fluid, or one of the brake calipers is stuck.

My quad has a foot brake like that and a hand brake, but the foot brake only brakes 1 tire. That tire has two brake calipers on it's roter, one to the foot break and one to the hand brake. So the hand brake is all 4 wheels, and the foot brake is 1 wheel. There is a master cylinder on my hand brake and another on my right rear tire (foot brake). If my foot brakes master cylinder runs low or springs a leak, my pedal goes through the floor like in your pic but I'll still have brakes on the hand brake. It's like a double/emergency braking system. If my hand brake goes out I can down shift (engine brake) and lay on that foot brake to brake and not hit a tree...
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Fuel wise, it's likely carburetor being old. The carburetor should have a primer on it that injects gas into the carb when you press/pull it. Those generally have a diaphragm in them (rubber) that tends to go bad over time and cause a fuel leak.

If you can find a diagram of the carburetor that would help emensely.

Really though, you should take the whole carburetor off, take it apart and give it a bath in carb cleaner (no plastic/rubber in there) and clean all the jets out.

Also, they make rebuild kits for most carb's that come with all new jets, and pilot screw etc. See if you can find one. I typically just replace them all, easier and then I have spares.

Also, inspect the carb's vent hose and make sure it's intact and not clogged. Check the fuel line too, for damage/rot.

You should also remove the gas tank and clean it out. Take the petcock off and inspect the filters and valve, then clean the whole gas tank out so there's no dirt/bad gas in there.

Then check/change the spark plug(s). Before running it though, I'd check the valve clearance on the valves, guides on that online.

Once all that's done, it should run good and not leak gas.

Then you can address your other issues.

Pending how old it is, there are some parts I might replace just because:

• Voltage Regulator
  
• Starter Solenoid
  
• Starter
  
  If the current ones are working, call them spares.
  
  Starters and solenoids are cheap, voltage regulators OEM are expensive, but you can buy a few after markets pretty cheap to have spares.
  
  Tool wise, I can recommend at least the following:
  
  
• Wire/Brushes
  
• Jack/Lift
  
• Compression Tester
  
• [Feeler Guages] (https://www.amazon.com/gp/product/B000BYGIR4) You want these to go from at least .002 to .014 inches
  
• [Caliper] (https://www.amazon.com/Neiko-01407A-Electronic-Digital-Caliper/dp/B000GSLKIW/) If you need to measure float height, you want to be accurate.
  
• Angle Finder Useful for float height, as most carbs need to be held at a specific angle for accurate float height adjustment.
  
  Then your typical ratchet sets, air tools, impact guns, etc.
  
  And socket extensions (long ones) because getting to some things is a royal pita.

They remind me of tip cleaners used for welding/carburetors.

https://www.amazon.com/dp/B076HMF3DM/ref=sspa_dk_detail_1?psc=1&pd_rd_i=B076HMF3DM&pf_rd_m=ATVPDKIKX0DER&pf_rd_p=f52e26da-1287-4616-824b-efc564ff75a4&pf_rd_r=4VXEYMT6PJFT4FAHBYRK&pd_rd_wg=VYUU5&pf_rd_s=desktop-dp-sims&pf_rd_t=40701&pd_rd_w=ihA2C&pf_rd_i=desktop-dp-sims&pd_rd_r=42fefff4-b975-11e8-be59-db2f2d1fb2cc

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Same thing.

https://www.amazon.com/Carburetor-Cleaner-Torch-Welder-Motorcycle/dp/B076HMF3DM