1) Generally any viral infection that runs down my immune system can create a cold sore. A flu, a bad cold or covid.
2) Sometimes just stress and/or being sleep deprived will do it, but there usually needs to be at least 3-4 months since the last outbreak for that to happen.
3) You're lucky if Abreva works for you. For me, the creams do almost nothing. The only thing that seems to help is Valaciclovir oral tablets. If taken early enough, they can prevent the outbreak or make it a very small cold sore.
So, in summary, I don't think there's anything special about covid with regard to cold sores. It breaks me out about the same as any other virus that runs me down / gives me a few days of fever.
You should also probably check out Valaciclovir, as it sounds like your immune system does quite well against cold sores, so the extra boost from a more powerful antiviral might completely stop your outbreaks.