If you cut down the trees and sequester the carbon elsewhere, you can sequester more, but just establishing new forests all over would still increase the amount of carbon cycling in those local ecosystems, and get it out of the atmosphere. Both is probably best.

I don't have any figures, but on the assumption that few processes are 100% efficient I'd guess you wouldn't get complete decomposition before the fallen tree is buried. You'd probably get a reasonable, if inefficient, removal of carbon from the atmosphere. Probably better than nothing if you couldn't figure out a way of using the wood more effectively.

Hm, as far as I remember that was the way CO2 was turned into fossil fuel, depositing layers over layers of vegetation...

I believe that accumulation happened before microorganisms had evolved to digest cellulose (?). If we can ever produce a surplus of electricity, we would ideally sequester carbon in a place other than the bottom of the ocean, creating a giant Earth-battery. Although it would be ironic if "carbon sequestration" meant heaping plastic in landfills.

Probably a really bad idea but what about chemical treatment to prevent rot.

That would negate the advantage of not requiring extra effort/energy. And decomposition of organic material is required for the rainforest ecosystem to work.

And I'd go with radiation or heating for sterilization.

or you know, maybe just using the new "easy" tech of lignin removel through oxidation. But you're right, we can't excessively clean the forest of organic "waste".