Catholics, if you struggle with a mortal sin, does it make you doubt the sincerity of your repentance?

No, not really. At the risk of sounding glib (there's gotta be a special place in hell for "bumpersticker theologians"), I would suggest that struggle would be evidence of sincerity. If I kept repeating the same sin and it didn't bother me that I did, that might be a problem.

I'll take it a step further: certain saints had lifelong struggles with certain sins. The fact that these heroes had to keep fighting reassures me I'm not alone in my struggles.

No, struggle is a good sign,it's the lack of struggle and surrender ( with full concent of the will and succient knowledge of the gravity of the morally serious action or decision)

The Church's answer is jesus': Confess, repent and receive the absolution from the Apostles and their successors

repentance is a process beginning and continuing with God's grace and the cooperation of our decision

If you are sincere in your promise to try to stop then yes. If you go into sin acknowling what your doing is wrong and you think "I will just go to confession latter" thats a sin of predetermination.

No, when you confess you must be sorry for your actions and have the intent not to repeat them. That said, many will fall to the same sin. The defining portion of weather your penance is sincere or not is weather you intended to stop your sin, not weather you repeated it.