Welcome to the family elpato. Glad to hear your bringin the bike back to life, sad to see them just sitting in a garage. As far as being new to Kawi, there's a reason that the pre-gens, that's what we call the 88-07 model, were sold for 20 years. Cause they're a great little bike.
Cleaning the carbs should help solve some of your problems. Give her an oil change, I like Shell Rotella T-6 5w-40. Install an in-line fuel filter. You can get them at the local auto parts store, I think it's a 1/4 in line.
Oil in the airbox is common with these bikes. It gets pushed into the airbox at high RPM. I'ts no big deal, just watch your oil level once you get it running.
Also the bike won't shift into second unless you're moving, so I wouldn't worry about it until you get the bike running. Transmission failures are not common with theses bikes.
If you have anymore questions don't hesitate to ask, and once you get her shined up lets see some pics.
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Don't do something because it's easy,.. Do it because it's not!
If you aren't going forward, then you're falling behind.
"Drive it like you stole it"!!!