Is my powerbook battery dead or my mac's ability to charge broken?

That "Full Charge Capacity" of 10000 is totally bogus. That's as far as I needed to look.
I have seen that a number of times with failed batteries, reporting erroneous data.
Battery is

I think all those mean the same thing.

 

That looks like a classic dead battery to me. The fact that it recognizes the battery makes me think that the charging circuit is intact.

The RAM slot failure is common on the DDR-based 15" PowerBooks.

 

A battery issues is a battery issue, a ram failure is a ram failure. But the battery itself looks bad as said before.

I would suggest that the ram issue is with the lower slot which is a common issue with the 15" powerboook see more about it here: http://lowendmac.com/misc/07/0102.html

 

It's possible, but it's never worked.

Been tried before. The memory controller won't recognize over 1GB of ram per slot.

 

It'll read a 1GB stick. That's how you can get 2GB with these Macs although Apple always said 1GB was the max. There's a resoldering fix, but it's up to you if you want to attempt that. Failure will kill the logicboard.

My son has a 15" PowerBook and it came with the lower slot failure. I just left it alone because he transitioned from a 1Ghz TiBook with 1GB of ram so he doesn't know any different. Him using the computer for as long as it lasts is more important to me then trying to fix a problem he doesn't care about and potentially ruining his computer.

It's an individual choice.

A new logicboard will fix the problem though, assuming the new logicboard doesn't also have the ram slot failure.

 

If it's the DLSD version, I doubt it. If it's the standard 1.67 it might. Apple changed the connector on the LCVD cable with the DLSD models.

If you can find a good logicboard it's not a hard fix. I've changed three myself and I've completely removed the guts of a 12" PB in order to replace the LCD.

A fair price would be anywhere from $10-35 USD. I wouldn't pay more than that.

 

Find out the model number of your Mac. Mactracker is a good app for that. Then do a Google search like this: ebay.com PowerBook+G4 A1013 logicboard

I'm using my model number, which is for a 1Ghz 17" AlBook, just put in your model number. If you don't get any good results, change the logicboard search term to motherboard. Alternatively you might use ebay.ca instead of ebay.com to make it local to you.

You may come up with some auctions that have already sold. That's ok, pay attention to ebay's alternate suggestions. That can lead you to stuff you want.

Using Google to search eBay is always the better bet over eBay's own search.

 

Odd. I don't have an answer for that one.

Good luck on the hunt for a Quad. My best advice on that would be to make sure the LCS (Liquid Cooling System) is in good repair. Nothing has killed 2.7 Duals and 2.5 Quads more over the years than a leaking LCS.

 

I used to have a 1.67GHz, NON-HiRes version of PowerBook and the bottom RAM slot was iffy. Sometimes it recognized the RAM, sometimes it did not. Trust me, the slot it "on it's way out."

As for resoldering, is this not a case where a "reflow" may work? I know this is normally only recommended for "dead" components, but it may be worth a shot if he really wants the slot back.