1. I'd use copper foil to repair the wide trace - you can do a neater looking job of it than with wire. The other traces can be repaired with appropriate thickness strands of tinned copper wire. Tin all the traces you need to repair first while yo've got good access, before doing any other repair work.
2,3. Yes, PCB lacquer would be good, applied *AFTER* all other repair work is complete and the board has been de-fluxed. You will need to glue some of the repair wires down anyway and in addition to protecting the tracks, the lacquer will also help bond the repairs in place. If you are going to put this project aside, a light film of liquid Rosin R type flux (not RMA or RA, or gel flux) or pure 'white' Rosin dissolved in alcohol applied locally to the tracks you have cleaned to bare metal will dry to a coating that retains solderability for months or even years, but you'll need to wash it off again with alcohol of flux remover before lacquering it.
4. I wouldn't use a board mounted coin cell - there's no guarantee it will even work with your board as the original battery was 3.6V, and if it does work, you may find it needs a new CR2032 every three moths!
I don't know if the four pads I see the other side of C31 from the 14069 footprint are for an off-board clock battery - if they are an off-board 3xAA holder with a series diode to provide anti-charging protection and drop the voltage a bit would be the best option. Alkaline AA batteries will keep the CMOS data and RTC running for *YEARS* and with their holder Velcroed to the case, its very easy to keep any potential leakage well away from the circuit boards. Worst case, you could reinstall a new 3.6V NiCd, but if yo end up doing that, fit it after lacquering the board, stick a rectangle of Kapton tape over the battery footprint extending a half inch outside it, , smear a little lacquer over the tape edges to seal any small gaps where the tape doesn't lie perfectly flat due to tracks and/or repair work, then pierce the battery mounting holes with a needle so the pins are a tight push fit through the tape before you solder them. With a drop of lacquer applied after soldering where each pin goes through the tape to seal them, the board will be well protected against any future leakage.