Bearing in mind that I haven't paid much attention to tactical repair since v1.0 or so:

You recover the repair rating in HP every 'tick' of the engagement, with 'ticks' being timesteps of a duration I don't know which is much shorter than the period of the firing cycle of any of the weapons.

As far as I know, the maximum damage recovered per fight is [damage sustained during this fight] + [max HP] - [HP at start of fight]. Last I paid attention to it, you could literally recover hundreds of HP per fight, or even per ship.

No, but as the battle viewer is there to be pretty rather than to be an accurate reflection of what is actually going on in the battle simulation and as repair appears to be nearly continuously active as long as a ship is both damaged and within firing range of an opponent, notifications would probably not be worth anything anyways. If you want, you can select a damaged ship with tactical repair and watch its HP recover rapidly as long as it's actively engaged with an opponent.

My experience is that very few individual ships are saved by tactical repair, but that the fleet as a whole is much less damaged after a rough battle.  This is especially useful if you are in a scenario where your fleet will be battling multiple enemy fleets with few turns available for repair delays.  You can also game the system by attacking a weak ship or fleet and getting essentially free repairs.

On the amount of HP restored by tactical repair in combat? No, not that I'm aware of, aside from those limits imposed by time constraints and maximum ship HP.

I agree with erischild's view that tactical repair is unlikely to save any given ship, except maybe in an edge case. Like erischild, I feel that tactical repair is more useful for keeping the fleet in decent shape after a battle than for its ability to actually save ships. I will add that a part of this is due to the fact that a damaged ship with tactical repair will start to recover HP even before its defenses fail, which means that if the ship's defenses are good enough then a ship that entered battle at half HP can be at full HP by the time defenses fail and so taking that half-HP ship into battle is effectively the same as taking a full-HP ship into battle (remember, however, that this is something which depends heavily on the relative strength of your defenses to their weapons; sometimes you can only recover a few HP before defenses fail, other times you can recover hundreds). The other part is of course that ships will recover some of the HP they lose after being hit and so effectively take less damage during the battle.

It will reduce the total manufacturing cost by comparison to filling that space with standard defenses, yes. By late game, it may even be reducing the overall manufacturing cost of the ship by a few hundred points. However, relative to the total cost of the ship this is fairly negligible, and I would not expect it to reduce construction time significantly at any decent shipyard.