Zardnaar
Deity
The idea of Moscow being 3-rd Rome is from 16-th century, before the Romanov dynasty was founded.
After the fall of Constantinople, the idea was that Moscow tsardom is a "successor state" of Byzantine Empire, because it was largest Eastern Orthodoxy state and inherited many cultural things from there, such as the alphabet. Also, Ivan III dynastic marriage, which you mentioned and the general resurgence of Moscow state (it got rid of Mongol yoke about that time and became much more powerful).
It has pretty much nothing to do with Romanovs, rather with Moscow as the city and Moscow-based state, which was essentially born right after the fall of 2-nd Rome and survived to this day
I know but her bloodline is still present in the modern Romanov pretenders afaik.