Okay, let's go down your list one at a time:
Clutch Actuator - If you are missing the clutch ball retainer, you will need to find one from a parts bike or through Honda. I wouldn't say that it's bad, but it won't operate properly without it. Sticking a few loose balls in there won't work, so don't waste your time. Be sure that you also have the single #10 ball that fits in the back of the clutch lifter. It is often lost or overlooked when the assembly is taken apart.
Tube - Frankly, I can't understand why you would want to patch a tube and risk your life for the sake of $10. Buy a new one - they're a lot cheaper than a hospital stay. While you're at it make sure that the tire is not damaged. If it is, buy a new one of those, too.
Carbs. - Does the fuel leak out of the overflow tubes or the sides of the bowels? Either way it sounds like you should check the float adjustment first, but you may have additional gasket-related troubles. If fuel is overflowing, it's possible that it's making it's way into the crankcase. Check your oil to see if it has an odor of gasoline. If it does, change it. Gas will always win over oil and you will destroy the engine running it that way. Make sure to shut the fuel petcock off whenever the bike is not in use. These bikes are known for fuel leaking into the crankcase, particularly if the overflow tubes are clogged - as often they are.
Backfiring - It could be a timing issue or it also could be a valve issue. Precise ignition timing is critical on these engines, so start there. Make sure you follow the instructions found in the owner's manual or shop manual step by step and word for word. Also, make sure that the points are clean and not burnt or pitted. If they are, dress or replace them. If that doesn't solve the backfiring problem, run a compression test. Lastly, be sure to adjust the cam chain tension as per the instructions in the manual.
Take a look at the posts in the "Non-SOHC" section of this forum. There are many strings that deal with Honda 360 twins.
Best of luck,