It looks like you're doing great!
It's fine to clean the cover off, but I don't think water is going to cut it. I usually just wipe mine off with a rag or scrape it with a screwdriver if stuff is really stuck on there.
Since you're working on it (and it's always more fun to work on clean equipment), I would spray the dirty parts down with some degreaser (Mean Green from dollar general works awesome) then wash everything off with some water.
Here are the steps that I do when going through a new (to me) saw.
1) Clean all of the crud off from it that I can easily clean with compressed air and a rag so I can see what's going on.
2) Pull the spark plug to check for foreign objects. If you see metal or anything that doesn't look like it's just carbon (burned on oil deposits), you'll have to pull the whole engine apart to clean it out. You'll probably want to do that eventually either way, but I wouldn't if everything is clear.
3) Put the spark plug in the boot (the black rubber thing that snaps onto the spark plug) and set the metal part of the plug on the outside of the cylinder (big metal thing that the spark plug threads into) to ground it. Turn on the power switch and pull it over. Don't touch the spark plug when you do this or you might get a jolt (my pops thought that was as funny as hell). You should see a spark at the end of the plug.
4) If you saw a spark, pour a little bit of gas and oil mix into the spark plug hole, thread the plug in and try to start it. If it starts up, put some fresh gas in the tank, pour a little more gas in the spark plug hole and see if it'll stay running. If it won't, it isn't a big deal, but you might need to clean your carb. If it won't even pop, then check the cylinder for scoring.
Hope that helps,
Eric