I also have caliper along with a bunch of other stuff.
http://suzukisavage.com/cgi-bin/YaBB.pl?num=1466172778/0Before you start mucking around with the carb, I would oil the cylinder, make sure you have a good battery and see what happens. You might be surprised. I always like to see how things go before I tear into anything. Then I work through potential issues one step at a time.
For the neutral issue, roll the bike back and forth while working the lever. This will get the gears to rotate a little and hopefully allow you to find neutral. DO NOT mash on the shift lever. All that will do is bend the shift linkage and it will eventually break. It may already be bent and contributing to the problem. You might want to take a look at that too. You don't want it to break while you're out on the road and have to ride home in first or second gear. I found this out the hard way.
EDIT: Just realized I missed Varsy's post about the neutral issue. Do what we said.