congrats on your purchase ! I'm sure you'll love your first AGA amiga.
I know about the furry-battery issue, is it possible to run OK with the battery removed like the older Amigas sans battery backed clock?
yes.
From what I recall the A4000 has no built-in flicker fixer - will I be able to boot and use WB "out of the box" with a VGA display? Do you all recommend a flicker fixer or trying to find a C= 1960 or NEC multiscan?
You can use WB in multiscan mode with a VGA display. However first you must use some kind of 15 KHz compatible device to change the screenmode from workbench (a TV will do if you have a modulator ?) I recommend a scandoubler. Old multiscan monitors are approaching end of their lifespan so it will be a bad purchase.
Can I just add a CD-ROM (or even DVD!) drive as a slave on the built-in IDE interface?
yes, but you need some software like IDEFIX.
Any other warnings/tips/tricks for the A4000 or AGA Amigas in general (I las had an A600, and only briefly anything AGA) would be appreciated.
Ekectrolyte caps at A4000 audio filter are known to leak, so if you can, change them or have them changed. Ram slots are notorious for weak plastic clips. Don't mess with ram unless you really have to. Finally, while a 030 is fine for simple retro pleasure, I highly recommend getting an accelerator with some RAM on it to get machines true potential.