Definitely agreed on the martyrdom ability. It also doesn't seem to fit the advent as well. I can understand the whole "Give your life for the advent", but it seems a sort of pointless gesture, even with the level of devotion of a seeker crew. And with lingering presence, the advents sight specialty is already covered. Perhaps something that affects culture spread, either you own or an adversaries. It doesn't seem unreasonable that seekers could be outfitted with broadcasting equipment, or some such thing.
That seems reasonable. Perhaps something that lowers resistance to culture spread - then it's only effective if you've already got some spread over or approaching the enemy planet in question, especially if they've got broadcast centres built at said planet.
And as for Jump Degredation, I'd like to see that be a bit more useful too. As it is it's only useful for moving a defending fleet an almost infinitesimal distance into a marginally better position. That, or to start moving the rest of an outnumbered attacking fleet towards a phase lane in order to retreat (leaving the overseer behind in the process, but that is after all their job).