My opinions on the OP, most of which I disagree with:
1> Starbases should ALL move, but very slowly. Think of them as being towed from point A to point B, or overcharging their "station-keeping" engines (which they'd have to have to maintain orbit). Concept aside, they simply NEED to be a little mobile, or else they become tactically useless without an insane range increase. And there's no comparison to capships; bases are SLOW, and take forever to cross a gravity well. Within combat, their movement is negligible.
2> No "Easy Button", please, but I would say that more things should be capable of spotting mines. Starbases, for instance, should duplicate the scouts' mine-detection ability. This'd allow systems to automatically clear out enemy minefields over time, using only their existing hangars and such. Of course, with the announced change to scout AIs, this isn't really necessary; leave one scout behind in each seized well, and it'll clear out the mines itself (or use the hangar fighters to do so). I fully expect most players to have a few small, dedicated "minesweeper" squadrons with a couple scouts and a handful of flak frigates, solely to quickly clear out enemy minefields in newly-seized systems without any micromanagement.
3> Definitely no to multiple starbases; you'd very quickly reach the point where a system is utterly impregnable. As for the range increase, I'd be for that IF it only applied to a single weapon type. Say, missiles; the beams and guns on each SB would have normal ranges, but the missile banks would reach further (some sort of "capital missile" that wouldn't fit on a normal ship). This'd be enough to guarantee the base couldn't be bypassed, without creating the sort of range advantage that'd prevent an enemy fleet from ever entering the base's range in the first place. (Imagine placing the base at the far end of the grav well, but with enough range to reach all the way across, and figure out how much damage it'd to do a fleet trying to reach it.) Problem is, this change isn't really needed any more; fighters and bombers already cover the entire well, so just put more of those on your SB if you're worried about bypassing. And note the change in the patch notes about "rebalancing" SB hangars; if a starbase can now have a dozen or more bomber squadrons with ease, then nowhere in the grav well is safe for enemies.
4> Yes, it should, or else you'd only ever need one, and it'd become a much more high-priority target. If the cost of the constructor was raised tremendously then maybe it could be balanced this way. Also, if the constructor had to sit there and actually construct the SB for two minutes or more, and the enemy pops in and destroys it while the SB is half-finished, then what happens? Under the current system, there's no way for a base to be stuck in that half-finished state; you either destroy the constructor before it starts (in which case the builder hasn't spent resources), or you try destroying the constructing base itself.
5> Incomplete starbases should not fire, period, IMO. But if they do, the same rules should apply in enemy wells as in friendly. The easiest way is to scale damage by completion percentage; a 50% done base does half damage, etc. Since building in enemy wells now takes considerably longer than in friendly, it'll spend far more time in that incomplete state.
6> No z-axis, please. Adding another element to the game's strategy should not be done lightly, and starbases are not a good reason to do so; you'd have to add a third dimension to every other part of the game. As long as the bases can move a little (#1), there's no need for up/down placement, since the blind spot wouldn't be permanent.
7> As long as you don't get anything back for a destroyed module, I'd be all for this, although the balance could be a bit funny for modules that duplicate structures (trade port).
