jheronimus wrote:1) can I try to start the system before replacing the battery? I'd like to know if it's working before I deal with it.
Yes, if the leakage has not affected anything yes and the board is working it should start up without issues.
jheronimus wrote:2) do I really need to change the battery? Not sure what's it used for on a system like this besides the time/date/BIOS settings
Well, at the very least you should remove the leaking battery. If you can live with having to re-enter BIOS settings each time you power up, the system should work without the battery (some boards refuse to POST without battery, but I think that's mostly when an RTC chip is used).
jheronimus wrote:3) is it doable? I don't have any soldering skills, but I have a friend with soldering iron. He can do this if I can show him what needs to be done and I can source the battery.
Removing the battery is a veary easy job. Just apply some heat to the joints and rock the battery a bit to crack the glue blob. It should pop right away (you can add fresh solder if you are having trouble with the joins). As for replacing it, the recommended thing to do is buy/build a non-rechargeable battery pack out of AA batteries or similar and connect it to the 4-pin external battery header located near the barrel (there are several threads here on how to do it). You could also replace it with a new barrel, but at some point it will leak again.
jheronimus wrote:4) what should I use to clean the traces of the leakage?
As the battery electrolyte is alkaline, an acid substance such as vinegar or lemon juice can be used to neutralize the corrosion. Note that this won't fix damage if the corrosion has gone too deep, but I can't tell how much damage (if any) there is since the battery kind of blocks the view.