How is molecular weight and thermal stability related?

Answer 1

They are not related. Thermal stability depends much more on the nature of the bonds in the compound, not the molecular weight.

Boiling and melting points do generally reflect molecular weight (both go up with increasing mass), but thermal stability usually refers to chemical decomposition rather than physical changes such as melting or boiling.
There are very light and very stable compounds (carbon dioxide is one), and there are very big and stable compounds (polymers), and there are very light unstable compounds (hydrogen gas) and very heavy unstable compounds (TNT isn't huge, but fairly large...).