Yes, it can. You should go to a doctor and get tested. I know it scares you, but it will be better to have a definitive answer.
The burning feel can be psychological, since you are very worried about it. There are several other conditions you could have, like acne or an allergy for example. Being anxious doesn't help at all! Calm down and remember herpes and all the other conditions are treatable.
You can't spread your cold sore to your genital area, although very rarely you can spread it to someone else's during oral. You can get info about that on the std forum, so I won't comment again on this thread.
Herpes is a fragile virus outside the skin so doesn't travel. People get it on their lips from kissing which is why many people have it, including babies in older times when mothers allowed strangers to kiss their babies.