Guillaume:
Rule 37 answers the question: when must an undercall be adjusted to a full call and when might a player be allowed to leave the undercall in and fold. In all cases either heads-up or facing the opening bet an undercall must be made a full call.
In other situations, at TDs discretion, the player might be allowed to leave the undercall in and fold. The rule does not guarantee that such players always have the right to leave the underbet in and fold, it rather leaves it to TD discretion, keeping in mind that an important responsibility of players is to follow the action (Rule 2). Some TDs rarely recognize undercalls and simply require all betting action be adjusted to the proper amount. Other TDs will allow the player to leave the bet and fold in cases of "gross misunderstanding". Yet others may allow "leaving and folding" more frequently. In the past, the TDA has debated the subject of gross misunderstanding and attempted to set a percentage for that, but nothing was agreed to.
In the specific case you cite, while C technically was not calling the opening bet, 2000 seems like a small absolute difference in the bet amount. I would tend almost always to require a player to bring their bet up to a full call in such situations and would not allow C to leave the 3k in and fold.
That's a long answer to your question, the short answer is it's your discretion but (to your sentence in caps) if you do allow C to fold, yes the 3000 must remain in the pot because this was a bet in turn and chips in the pot in turn remain in the pot.