the lever on the left side...it's probably the arm for the coaster brake. without it, you won't have any brakes and your axle could possibly rotate backwards when you try to stop. This could cause the wheel bearings to come loose.
it's easy to put back on because it only fits onto the hub 2 ways. one way will be right, the other way it wil be upside down. it has flat spots in it that line up with flat spots on the axle nut. (this is the nut that holds the bearings in...not sure of the correct name for it) The brake arm slides onto the axle nut with the 2 flat spots, and then a regular nut goes on over the beake arm to hold it tight. If you put the regular nut back on the axle before you put the brake arm on, you'll never get it to go back on. (pics of what your trying to figure out would help.) I know that you had to take the brake arm off to get your sprocket on and i'm thinking that you put the regular nut back on before you put the brake arm back on. In some instances you have to slightly bend the brake arm so it clears the bolts on the sprocket....just an f.y.i.
If you put a regular washer under the fuel on/off fitting, it will leak gas big time.You need to use a fiber washer which will take up the same area as a regular washer, and it will make it dso the lever isn't hitting trhe frame. your kit shoudl have came with a red fiber washer for this. some teflon tape or gas fitting sealant on the threads will also help seal it. these are cheap valves and they are known to leak.
as for the cdi wire connectors, do yourself a favor and cut off all the connectors. (also cut off the butt connector that you installed.) solder all wire connections and cover them with heat shrink. this will give you more reliable connections, and they will flow a little more juice. those plug in connectors and butt connectors will work, but they can rattle loose, pull apart, and they are give a lot of resistance. soldering your wires and heat shrinking them also makes the wiring look a lot cleaner and more professional in my opinion. I have no idea why i read that so many people have to ground one of the wires to the frame. I have built 3 bikes and have never had to do that. Just hook the wires up as the instructions say and it will be fine. I can't tell you how to hook them up because the wires on my bikes could be different colors than yours and i don't want to add any confusion. all i know is that i hooked my wires up (for example red to red, blue to blue and black to black) on all my bikes and they all work perfect. The colors of your wires may be different so i don't know what to tell you. the colors of the wires don't matter, but what does matter is where they go. I am not a wiring expert so i can't really help you out there.
actually to cure the exhaust rubbing on the cranks, you can bend the pipe (not the muffler). You shouldn't need to re-position your motor and actually that probably won't help you because you will only be able to move the motor forward or back, not side to side. but i see what your saying...if you move the motor forward a little it will raise the pipe and it might gove you some more clearance. personally if it were me, i'd just take the pipe off, and put a small dent in it with a ball peen hammer where the crank is rubbing.
you can also take the pipe off the motor and put it in a vice. heat the pipe up where you want to bend it stick a steel bar into the pipe and bend it. you can only bend the pipe itself, not the muffler. whatever you do, don't try to bend the pipe while it's on the motor because you will break the exhaust mounting studs. putting a small dent in the pipe might be your easiest solution tho.
when you install the slide in your carb after you hook up the throttle cable, make sure you put it in the right way. if you look at the slide when you take it out, you will see that it has 2 slots cut in the sides of it. the short slot goes to the left side where the idle screw is. the long slot gets lined up with a pin that is inside the carb body. if you put the slide in wrong it will either be at wide open throttle all the time, the slide can stick and hang up and you wil not be able to adjust the idle. also make sure that you drill a small hole in the handle bars where the throttle handle goes on, there is a pin in the throttle assembly and that pin needs to go into a locating hole (that you need to drill in the handlebars) to keep the throttle assembly from rotating around on the handlebars.
that part that you havem that you have no idea what it is... I'm pretty sure that it's a clutch removal tool. none of my engine kits came with one but i'm pretty sure that's what it is.