Iodine only protects the thyroid against radioactive isotopes of iodine. That's interesting after a nuclear explosion or a major power plant accident because Iodine-131 is a significant product of them, but it doesn't do anything otherwise.

> How long would someone need to be cryogenically frozen for cosmic radiation to be a significant threat?

A short-term dose of ~100 mSv is the lowest amount where we are sure it increases the cancer risk. If we assume accumulated dose during freezing acts like a short-term dose then we need ~200-300 years to get there. Damage from the freezing/thawing process is probably still your main concern here. If we look at doses so high that they can kill you short-term then we need over 1 Sv, or thousands of years. This is assuming no special shielding in any way, and it's also ignoring terrestrial radiation sources. Normally most of the radiation dose comes from that part and people get something like 2-3 mSv/year, so we would reach 100 mSv after 30-50 years or so and over 1 Sv after a few centuries.