- I'm having a real tough time finding some 220gr cast lead .30 cal bullets sized above .308. My Krags barrel slugs to .311, so I guess I need .311 or .312.

Did you slug the throat (leade)?  If not, verify the groove dia there is also .311"  Also, get a sample .311" or .312" bullet & seat them is a case to make a dummy round.  Verify the larger bullet, seated in a case, will easily chamber. \

Get a copy of a good loading manual.  I prefer the Lyman 49 and it covers both jacketed & cast bullet loading techniques & data with a variety of components.

- What powder is good for a mild load for punching paper?
Any of the 4895 powders work.  For cast bullets, consider 2400.

I've read even some medium loads can crack the bolt on a Krag.  I haven't heard this but I suppose anything can happen.  Have a competent gunsmith, familiar with Krag actions, check yours if you have any doubts about it.

- Do I need to trim rifle brass in a bolt action after every firing?

Maybe not after every firing but perhaps every 2-3 firings.  The Lee case trimmer system is the cheapest & easiest system to use.  Measure cases after each firing to make sure they do not exceed max case length.

- Remington or Winchester, which .30-40 Krag bass is better? Apparently some prefer Winchester but Remington is all Midway USA has in stock.

I prefer Winchester but Remington is fine as well.