Buy the clymer/haynes shop manual for the bike and dig in. The Honda manual is great too, but it assumes you know the basics. The third party manuals give more practical advice. Also ask on forums like this one if you're still stuck.
But yeah, doing it yourself is the best thing.
The chain slack is listed in the owner's manual, usually it's about an inch to an inch and a half at the loosest point. Usually given as 25-35mm. You measure by pushing up in the middle of the part that hangs down. Hold a ruler behind the chain. Measure on several parts of the chain (rotate wheel) because sometimes chains have uneven spots. That inch of slack should be at the tightest point, but if the difference between tightest and loosest is pretty big you need a new chain.
This isn't a carb sync issue. Carb sync manifests as extra vibration and slightly poorer performance.
It doesn't matter what the outside of the carbs looks like. Like some women claim, it's what's inside that counts.
Choke richens the mixture, and when you got the bike it was always on and you said it ran fine. Now that it's off, why would you think that it's rich? If anything it might be lean.
You say that the lights shut off too. That suggests it's an electrical issue, not carbs. I've had this happen, and it was dirty battery terminals. Just take a file or some sandpaper and clean the battery terminals and the cables that attach to them. Make them shine.
A friend also had it happen, and his main fuse was similarly dirty.
Bad battery may or may not contribute as well. If it's really bad it could be internally shorted and may drop the voltage too low sometimes. The battery is a crucial part of a bike's electrical system and it keeps the voltage in the needed range (>10 volts for the computer to work). You'll have a hard time getting $2200 for a bike that needs a jump. Buy a new battery.
I'm sure a new choke cable would be cheap, too. If not, you can fashion some sort of rope or wire you can tug on to open the choke directly. You should not mess with the mixture screws each time you start. Try explaining that to a buyer.
BTW, there's a mixture screw (one for each carb) and an idle screw (just one).