Special actions by minions

If a vampire/minion card does not specify otherwise, can that action be performed multiple times during my minion phase?

Can the medic burn the edge to give an anarch 1 blood, then another minion bleed to gain the edge, and the medic burn the edge again to give an anarch 1 blood? Can the same anarch get a second blood, or does it have to be a different anarch each time?

Thanks in advance.

The answer is: It depends.

See the Rulebook, chapter 4.3 "Minion Phase", at the end of the part on "Types of Actions":

"A minion cannot perform each action via the same copy of a card in play (including from the minion’s own card text) more than once each turn, even if they unlock."

This means that different minions may use an action via the same copy of a card in play.
An example: Different of your minions may attempt the political action to vote away an Anarch Revolt.

The same minion may not take the action again.

However, your example for The Medic does not involve an action at all:
It is a special ability of that vampire, not an action.

The Medic cannot use his ability twice in a turn because it follows the "“During X, do Y” template. (See Rulebook, chapter 3, "Wording templates".)