"Cold sores" have nothing to do with colds. They're caused by the herpes virus. Assuming that you don't have herpes, what you've probably got is a canker sore, which looks similar but has different...
Yes. A cold sore is a type of virus and it doesn't matter if you have a cold or not. The cold will not cause the cold sore but your immune system might be weaker making you more susceptible to it.
To me, both are pretty bad, but a canker sore is always worst! Cold sores can hurt sometimes and are a pain to have on the lip, especially because of it being un-comftorable! Canker sores hurt very...