1) 4 is the maximum per the game programming. Once restored, the player can again be injured (poorly cured, etc.), or a different player can join the others in the hospital.
2) Stamina is very important. The more tired you are, the worse you play, you react to others movements, create more fouls, etc. Lower the % of stamina is more % of these things happening.
3) Regarding tactics, most of it is just using common sense. Less agressive tactics would see less red cards, whereas more offensive tactics tire the players more... etc. Just try different tactics till you find one you like, use what the higher level players use (since obviously works for most of them), or ask around in the chat room. No one here is a Guardiola or Mourinho, but it´s fun trying to find a good and consistent tactic.
4) Who said that offensive tactics give less injuries/red cards? I use 69% attack and always use pressure. And I´m recovering players as much as others. The game doesn´t pick it´s victims at random... we all suffer the same, just sometimes the streaks are nicer than others.

Also like to remind you, this game is free. But for those who pay (or use their free boosters), a VIP rank will provide free restaurations, etc. Also, you still have a low level doctor and masseur. Build up your Medical Center and improve workers, you´ll see that you can recover them faster, etc.
Good luck!