Mouthwash usually contains some kind of mild anti-bacterial chemical which may help kill off the bacteria in the cyst. It is better to buy a proper anti-bacterial cream suggested by the chemist as this is much more effective.
I have been reading around and saw that a strong mouthwash usually makes the cyst/pimple disappear, if you see this, any comments?
It looks like a cyst, not a wart. The second picture shows not eczema but just marks from scratching, perhaps from stress. For the bump, you're going to have to show it to someone to get a proper diagnosis, but if you're not sexually active (and, if I understand you right, have never been), then there's no rush.
For the itch, apply an itch cream like non-prescription Benadryl cream every time it itches, don't scratch, and it should go away.
Take care.
Dr. Rockoff