1. I suppose you can store pearls in a safety deposit box with a jar of water, but it seems to me the extra moisture would not be good for the important documents you are also probably storing there.
If you are just putting them in the box for a short time (for example, if you are traveling and don't want to take them with you), the dry air will not be a problem. It's long term storage that is an issue.
The way I see it, pearls that are in a safe deposit box are not being worn and enjoyed. Better to find a safe place to keep them at home.
2. I don't think it's humidity that causes tarnishing so much as exposure to air. Tarnish is oxidation. My silver jewelry tarnishes less quickly if kept in a case or drawer. You can buy little anti-tarnish paper squares to tuck into your jewelry box, too.
When the findings eventually tarnish, I use a Sunshine cloth. If the strand has silver spacers as well, I restring, taking the opportunity to clean the silver.
3. I don't think there is any silver cleaner that is safe for pearls.
4. Not an answer to this question, but I urge you to try restringing yourself! It is the least expensive way by far, and Pattye's slippery Serafil thread makes it a cinch to place the knots correctly.
4.