That's the fuel petcock. I just changed mine on my yamaha grizzly last week because it was leaking also. You can get it on amazon for $9

Yes that valve is easily replaced. Your local quad dealer may have one in stock or a quick search online should give you several sources and prices .

Thats some lean. I feel like something you did during the lift was not done correctly or a set of bushings on the top swing arm are crushed. My approach would be to pull the front wheels off and take measurements from the bolt holes that hold the swing arms in place to the edge of the hub and compare the numbers. If your sure this was not a problem before the lift and nothings bent then its a matter of back tracking and making sure you did not mix up the lift brackets. There are two for the front and one for the back on the single suspension rear swing arm type atvs. Is it possible that you mixed anything up ? lastly some frames have two holes for the upper shock mount on the front shocks. Are they both in the same spots ?

All you need to do is take the light green wire coming out of the engine on the left side and unplug it. Put a jumper from the frame (ground) to the light green wire on the harness side. Make sure the bike is in neutral and turn the key on. I bet the bike dash lights will come on and crank.

That is your start circuit relay. Should have two black wires, one brown and one light green. The brown should have 12v power with the key on. The light green wire is ground and comes from the lower left side of the motor where the shifter is. Light green and red wires go into the engine. They go to your neutral safety switch. The lever inside when in neutral grounds the light green wire so the relay now gets positive from the brown and negative (ground) from the neutral safety switch. Once the relay has power it connects the two black wires on the relay allowing the start button to work.