Restriction 1: It is a bonus action
As a bonus action...
You can only take one bonus action per turn, so you can use this ability at most that often.
Restriction 2: It only works on one creature at a time
...you designate one creature you can see within 60 feet of you as
the target of this feature.
...
It ends early if you designate a different creature.
This is saying that you can target one creature at a time. When you target another creature, the effect stops working on the first.
This may seem a bit ambiguous, but it has been backed up by Jeremy Crawford:
Q: Can I get a quick clarification on the Monster Slayer's ability
Slayer's Prey? Does the final sentence mean I can use it as many times
as I'd like per short or rest, but only on one creature at a time?
A: That's correct.
(emphasis mine)
You can use it as many times per rest as you want
This benefit lasts until you finish a short or long rest.
This means that if you target a creature with the ability but don't kill it or move it to another creature, the ability will still stop working when you take a rest.
With no other text indicating a restriction of any kind, it would be wrong to assume there is one. And, in fact, Jeremy Crawford also confirmed this per the above tweet by confirming that it works "as many times as I'd like per short or rest".
Specific Questions
What happens if the creature under the mark dies?
If the creature dies, the effect ends and you can, if you want on your turn, spend a bonus action to target a different creature like you did with the first creature.
Can you carry it (so to speak) from encounter to encounter?
If you don't kill the creature or otherwise make it ineligible to for Slayer's Prey, then the ability will keep active until you take a short or long rest, in which case it ends.
You add your constitution modifier to the maximum roll of the dice
As you have noted, the medic feat allows you to forgo the roll and
regain the maximum number of hit points the die can restore.
Well, to determine the maximum number of hit points the die can restore, we have to know mechanically how the die restores hit points. That procedure, as you have noted, is:
the player rolls the die and adds the character’s Constitution modifier to it. The character regains hit points equal to the total.
Since that procedure is how we determine how many hit points a hit die restores, we would take the maximum possible result of that procedure to get "the maximum number of hit points the die can restore". The maximum result of that procedure is determined by the maximum value of the die plus your constitution modifier.
Thus, you do not ignore your constitution modifier when using the medic feat.
Best Answer
It's limited by the need to actually eat the food.
Because Xanathar's was written entirely inside a Magic Circle Against Editing, the writers didn't think to say "You and up to five other creatures can eat the meal to regain 1 extra Hit Point, etc." But that's clearly what's happening.
What you're describing is each member of the party eating three complete meals in one hour. That's probably impossible, and even if it was possible, it's not clear that you'd get extra healing from it. If that worked, then since you can make six servings of food with Prepare Food, you could just as well eat all six servings yourself and get +6 HP per hit die. Nothing in the ability indicates that you can do that--it's a buff that applies to up to six creatures, not six buffs that you can distribute among creatures.