Do tier rewards take up slots? For example I have one that gives me a bonus to invading planets. But if it takes up a slot I am permanently weaker in ship-ship combat. Also the one that reduces energy regen timer by 10 seconds is good, but when I am about to level I want + 50 energy. I think it's taking up my a slot. How do I un-equip them, or make it so I only use them when I need them?

Tier rewards are zero space, so they do not take up any decks, do not count towards your supply limit, and none of them have any negative effects, so they cannot be un-equipped as they only provide benefits.

IF an item has no uninstall button (or it is greyed out), it takes up 0 decks.
Tier rewards from my experience DO NOT give any negative effects (no reduction in any stat)
Side note: Psionic Amplifier is not a tier reward but a mission reward, thus it has some size.

So I can only normally equip 3 reactors? I could of sworn I had 4 reactors and the Omicron Injector took up my fourth slot. (Maximum x number of reactors per ship)

If you hover over the reactor name information will show up showing the limit for that type of module, the size, the condition and the upkeep. For reactors the limit is 3 total on a ship.

I did not notice that, very cool. Thank you very much

You can have a maximum of 3 reactors and 4 relays.

As Jarsyl said above, plus two special relays. Unfortunately no special relays are available at the moment though. The F.A.T.H.E.R. device was a unique item that went on sale for Galaxy Points (Note: Galaxy points cost real money) for Father's Day 2010. There was a M.O.T.H.E.R. artifact as well for Mother's day, but that was a planet item instead.

Keep an eye open for special artifact sales like these on special days. There will likely be a C.H.R.I.S.T.M.A.S. or R.U.D.O.L.F. item or something along those lines in the future.