From the building of my Titan stopping
While you can start building the titan with the first research it will only build 25%. You need to research the remaining 3 fully for the titan to fully construct. I agree this isn't the most intuitive for new players to Rebellion but it's easily understood after you get it down the first time.
to the inability to use crew extractor in ice fields
You need enough anti-matter for the ability to work. More likely..., It's also possible a random event was affecting the world at the time which disabled non-passive abilities.
not getting the bonus fighters in hanger defense for completing the correct research
The fighter(s, only 1 per level extra) appear in the squad. You no longer get an additional squad like in the early versions of this game.
Tonight my deliverance engines stopped working completely and I couldn't get any more to be built
You can only build 1 for every four planets you own (this changed with Rebellion). The ability only works if you maintain the appropriate research labs.