I realize that both of these abilities can be used once at the start of the Repair/Replace step (and this "step" occurs after every battle). But I have some questions.
#1 --- Must the Mechanic or Medic be assigned to that specific battle in order to use his abilities?
The End of the Week steps only happen after all battles for the week have taken place, so battles are no longer a concern during the Repair/Replace step. The Commander just needs to be alive at the end of the week to apply the ability.
#2 --- Can these 2 abilities be used only once per day or for every battle where there are damage counters or wounded/stressed Commanders?
These abilities are once per week (you might be confusing day/week here, not sure) since the Repair/Replace step only happens once per week (again after all battles for the week have been resolved).
#3 --- Can these 2 abilities be used to negate damage counters carried over from previous days?
I don't see why not. There's nothing in the rules that says it has to be damage from the current week.
#4 --- If the Mechanic/Medic chooses not to participate in any battles that day, is he still allowed to repair/heal?
Again, I don't see why not. The rules for Mechanic and Medic don't specify that the Commander has to have battled that week.