Is there a weep hole in the bottom of the BB shell to let the water out? I wouldn't close the ends of the seat stays or chain stays. Sealing them up could seal in a little moisture, and result in hidden corrosion that you won't notice until it's too late to save the frame. If you want to push some grease up into the tubes, to protect the ends from corrosion where brazing or welding may have exposed some raw material, that's probably all you need to do, but leave enough opening to let condensation drain out.

You can use either grease, or anti-seize, on the threads. It's easier to just use the same grease that you used on the bearings. If you repack the bearings, every year or two, you shouldn't ever have to worry about seizing.

Any quality grease is fine. I use marine grease, or axle grease, for hubs/headsets/bottom brackets. I wouldn't use white lithium grease for this purpose, and you probably don't really "need" marine grease, but the additional moisture resistance provides some peace of mind.