real shame about that iMac 1.25Ghz. :/ did you not think about offering it up here first? a dodgy head phone jack is an easy walk in the park repair people with even basic soldering skills. quite a stupid reason to chuck the poor thing... (im actually looking for a 1.25Ghz iMac G4 too, to try and perform some CPU upgrades on one... so i would of gladly taken your machine even if there was more things wrong with it)
By the sounds of things you have a Rev A G3 BW, if you can post a picture of the spec label on the back (if thats fallen off, post a picture of the CMD ATA chip on the LogicBoard)
they are very very finicky machines (I own 3 of the damn things

)
first of all, they can be very picky about RAM, so I recommend you pull your memory out and put in back what it came with. and seeing how it works.
if your RAM sticks are even remotely dodgy they can bring a Rev A to its knees, I find running gauge Pros memory test thingy will usually tell me if I have a dodgy RAM stick.
PRAM batteries are not needed usually for G3 BWs to work (mine work fine without em)
going by how your tiger updates are being corrupted im hedging my bets on bad RAM/RAM the G3 BW does not like.
second its not hard to bust a DMM out or even a scope and check the voltage rails on the G3 BWs PSU and see how they are behaving.
its also worth mentioning its very easy to adapt an ATX PSU to work with the G3 BW.