You can go to any Sears and get a
screw extractor (you can also get them as most other places that sell tools) to remove the stuck screws. To use it, you drill a hole into the screw then push the extractor into it while turning counter clockwise. If you have a high torque drill, you can chuck the extractor into it, but make sure you switch it to the high torque, low speed setting.
As far as the neutral issue, it is easier to find neutral while the bike is moving. In fact, I am going riding in a couple hours at a track where I frequently find neutral in the same corner (really soft sand with a large sandy slope to climb immediately after it). Changing the oil might help a little. The drain bolt is on the bottom (it should be pretty obvious which one it is, but I am not familiar with the particular bike).
Sign up here and look through the parts schematics if you are in doubt. Somewhere near the fill hole, you should find a number either stamped or more likely molded into the case, it will probably be something like 650ml, that is the fluid capacity. I use BelRay 75wt in my machine, but if you do a search, you will find a couple hundred opinions about oil preferences.
For your steering, it might just be loose, but take it apart and take a look just in case, I'm not going to detail the process, but you have to get the bars out of the way, pull of the top clamp, then a big pair of vise grips should get you by without the special tool. While you are in there, re-grease the bearings.