The out of combat staying time for both of these abilities should be increased by about 30 seconds, maybe even 60 seconds for Fleet Support since it's such a low proc rate. The timer should be reset to the top at the end of combat as well.
I've seen them drop out of combat because the didn't get engaged in time to be flagged as in-combat maybe? Too many times I've seen them drop out in the middle of combat and I'm not sure if this is broken or working as intended.
I firmly agree. Shoot the SE often beam out even during combat or never materialize - this is especially when used in conjunction with the tactical initiative ability (bug reports submitted). Simply put the current timeout is far to short especially given the CD on the abilities. If I had it my way the SE would stay until KIA or dismissed or that the very least can loiter around for at least half the CD time for the ability.