Some say "yes, by a lot". In my experience a 10 seconds fetch rate on apps costs only a few percentages of battery per charge if the fetch is reasonably sane. Checking 75 podcast feeds with thousands of episodes every 10 minutes isn't sane.

I have tested and got better battery on Lineage/Clean ROMs as compared with stock as well. I put it down to bloat and spyware constantly running in the background with elevated priviledge. 🤷‍♂️

Yes, the optimization is setting up how often the app and phone should wake up and check with your matrix server if there's any new messages.