Hi
I am unsure what could have caused the itchy skin. Itchy skin is a common symptom and there is a good chance it is not related to the sex you had 10 days previous.
Using a condom will really reduce the chance of an STI passing to you and the oil you felt may have just been the lube that was on the condom.
As you are concerned about this and many STI do not show symptoms I think your plan of going for a test would be a good idea. The end of the window period for HIV and other blood tests like syphilis would be the 12 weeks like you have stated, but if you wanted to go earlier you could.
Going for a test at 6 weeks would cover gonorrhea and chlamydia, also a blood test at 6 weeks for syphilis and HIV would also pick up most cases of these. You could just wait 12 weeks before going for a test but if you find it is causing you to worry the 6 weeks test followed by a 12 week test could be an idea as well.
Let us know if you have any other questions or concerns.
Health Nurse