325 won't start

I am the proud owner of a used 2003 325i. It won't start. I go to crank it, but it does not turn over. The headlights flash, the dash lights up with Service Engine Soon, ELM, DSC-off ("Circle A"), the radio briefly shows radio station info then reverts to the clock although you can't hear anything, and when I turn the key back to off, there is a winding down sound between the radio/climate control column and the engine. I got a jump and it cranked right up, then drove with no problems until I turned it off, then had to jump it again. Last week, I removed the flashlight from the glove box; it illuminated briefly then died and would not illuminate again. After driving the car for about 30 minutes, the flashlight still does not light up. When I removed the cover over the battery in the trunk, I can see liquid down in the battery well, although I can see no signs of puncture or leakage from the top. Don't know if that's battery acid or rain. If rain, how did it get in there? I can see some water signs in one of two round openings which contain wires heading to the rear lights presumably. Will a battery replacement fix my problems? The negative terminal looks like it's stamped with 3023.