IPhone 4S logic board replacement

If you have the Apple Service Toolkit, that is the easiest way, if you don't... there aren't may programs out there really suited for this type of work.

If you are showing that the battery may be holding or attempting to obtain a charge, the battery probably will be OK to be used. I am sure that enough time has passed in where the battery has been completely dry. In many cases you could submerge the battery, dry it off and it will work. The batteries are pretty much air tight to begin with.

If a battery had a crack in it, or something of the sort you'd know.

The best thing to do is to attempt to get a charge out of it, if the battery holds one you are fine, if not then you may be able to go down to an Apple Store and ask if there is a way to view how much life is left in the battery.

Of course, this is assuming the battery hasn't been saturated in salt water either, if it has, you may just want to throw it away. Otherwise I think you will be safe to use it as long as it holds a charge that is.

All of your data is stored on the flash memory, which is one of the chips soldered on to the logic board. It's not really possible to swap the memory to a different board (perhaps if you had advanced electronics skills, but I wouldn't like the chances of success).

If there's a problem with your logic board, you can replace it with a new one, as detailed in that iFixit link. Assuming everything goes to plan, and you have no other issues with the phone, it will effectively be a new iPhone. If you have a backup of your old one, you should be able to restore your data during the set up process.