Asymptomatic rash in the penis is rarely a STD.
STDs usually cause pain, discharge or ulceration.
Unprotected sex is just a coincidence.
Your rash is likely due to either atopic dermatitis or balanitis circumscripta.
THis can be caused by reaction to latex condoms, detergents, duodorants, trauma, chronic infection.
These are benign conditions.
Usually corticosteroid ointment (hydrocortisone, desonide) or tacrolimus cream may be helpful.
If persistent surgical excision provides permanent cure.
Yes, this is not strange.
This rash can persistb for long time without any treatment.
The rash does not usually spread especially in cases of balanitis circumscripta.
A local skin biopsy along with clinical examination will clarify the issue.
You should see a dermatologist.
Tacrolimus cream 0.1% or hydrocortisone 2% may be tried.
You should use till the rash resolves or for atleast 2-3 weeks.
Before using these agents, it is better to rule out any bacterial or fungal infection (not STDs, but infections caused by 'skin bacteria').
You should not appy any cream till you see the dermatologist for proper evaluation.
This condition is limited to the affected area you mentioned.