Hi guys, Ill try to keep this as short but as detailed as I can.
So I have an 87 ls 650 that I purchased less than a year ago. Allegedly ran with no issues was just taking up space in someones garage. Got it all cleaned up and had it running great. Its always had a bit of trouble starting up. Nothing too serious just needs to turn over a few more times than normal. Since I had plenty of gas and a brand new battery I was pretty confident and saying it was a starter issue. Put about 5,000 miles with it running the way it was no issues at all.
So over the winter I replaced the air box with a cone filter. Put a straight pipe with no muffler/baffle on it. And I replaced the famous 'leaking head plug'. The black stopper like plug under the cylinder head cover. Started running a bit differently because of the new air flow. Didnt want to stay idle, lots of backfiring even a few flames sometimes. (this was all done during the winter in PA so 50 degrees or below) So my friend and I spent a good bit of time working on adjusting the air fuel mixture
side note*He has a 2006 Suzuki Boulevard that he turned into a bobber. He did it all himself thats the only reason i trust his judgement with my bike
*. We got it running well, but it needed to warm up before it would idle without the choke. (used second stage to start and run in for a while) Well recently the temp around here shot up to 85-90 degrees with (what feels like) 100% humidity. So on the way to work one morning the bike just shut off on me, and had trouble starting back up. Ever since then Ive noticed its been sputtering while Im riding, more when Im at lower RPMs but randomly throughout. And were talking sputtering like darn near stalling out, very noticeable. So I go to look over the bike yesterday and noticed the negative battery terminal is melted, and the bike wont even turn over let alone start up half the time. It will shut off when I press the ignition start up button and if I wiggle the negative battery cable around it turns back on. I can tell this becuase my neutral light acts as an indicator. Im not a mechanic I just use common knowledge and the internet. 90% of my bike knowledge comes from this site.
But at this point my assumption is I have a crappy started thats taking too much time to start up which would cause my negative battery connection to heat up and melt the lead terminal

. As far as the sputtering Im sort of lost...

So my questions are;
What are the chances the electrical problem and sputtering issue are related?
Where should I start as far as diagnosing these problems?
I know with the new less-restricted air flow that up jetting is somewhat crucial. But if thats the case why has it been running so well in the winter? Are heat and humidity really that big of a factor?
Any and all information/advice will help greatly. Also the kicker is that I need to get my bike inspected this week because Im taking the motorcycle safety course on the 4th. So the sooner I can figure this out the sooner I can get what I need to fix it. Thanks guys. Let me know if any pictures are needed or will help diagnose things a little easier.