Starbases do not culture flip. Just be careful though, as starbases outside your territory might become difficult to defend or upgrade when you get involved in a war.

One trick I like to use is to place a constructor next to every mining starbase owned by some other civilization.  Eventually they'll go to war and either the starbase will be destroyed or the civilization will be destroyed, and then I can grab the resource.

Hasn't ever been a problem.  I tend to grab and max out every economic resource I can find, research the economy boosting techs, trade with everyone, and build up my population as fast as I can so as to have more taxpayers.  I do occasionally lose a constructor when someone declares war on me, though.