On a vx Suzuki 800 will gas run through pedcock when not on bike

As it is a vx engine going into a vs frame Suzuki may have used a different wiring harness to allow for differences in engine layout and the electronics fitted.
Is the engine the same age as the frame it's going into? Age differences can also affect the wiring layout.
You will need a wiring diagram for your bike and a separate one for the replacement engine. You will need the layout of the engine sensors as well.

Put the clip in the middle. If the bike tends to show a hesitation when starting out from a stop. move the clip lower. This provides fuel sooner at the low rpm. Move the clip higher if the bike is running rich.

If it dies when being put in gear, it could very well be the safety wire on your kickstand. check and make sure the stand is up all the way

fuel,fuel filter, vapor lock, vented gas tank not venting,bad gas caps, dirty screen,maybe at pedcock,sounds like a fuel problem to me!

old gas, fuel filter in the tank, air filter clogged on carbs , sometimes suzuki likes to run on 93 octane , more responsive.

more than likely it filter if you let sit for long periods and same gas
ethenol produces water,after long periods, let me know ..

I think you are saying that the exhaust on your bike is rusted so it looks bad and the bike is noisy because the muffler is bad. You can get a brand new exhaust system at Babbitt's for about $830.00.
Google " motorcycle salvage suzuki " and you might find what you need there for a whole lot less.

have you removed the spak plugs an turn crankshaft to see if not hydro locked

YB16 (Yuasa) or you ask for a 12V 16Ah battery with the positive pole righthanded as you ride the bike.