GR,
I would definitely replace the battery. That DA probably has the original battery. When I bought mine last year, that was one of the first things I needed to do, and as inexpensive as they are, it's really just routine preventative maintenance to replace it.
And as I said earlier, a weak or dead PRAM battery can be the cause of many ills and strange behavior, and your computer seems to have it's share of issues!
Chris K