Welcome to the forum. I'll try to ease your fears.
Like many young people these days, you have grown up with an exaggerated view about HIV, how common it is, and heterosexual transmission. Here are some facts: Heterosexually aquired HIV is very uncommon in northern Europe, especially Scandinavia; on that basis alone, it is very unlikely your partner has HIV -- assuming he isn't bisexual or an injection drug user. Getting tattoos is probably not a significant risk; modern tattoo methods are very safe.
Second, HIV is harder to transmit than you seem to believe. When someone has HIV, the average transmission risk for vaginal sex is somewhere around 1 in 1,000. (Many wives and husbands of infected persons never catch HIV!) And that's for completely unprotected sex. There is no signficant transmission risk with the minor and brief unprotected exposure you had.
For those reasons, I agree with your doctor: having no current STDs, and with neither of you at risk for HIV, you probably can safely have unprotected sex. There are some STDs either of you could still have, such as hman papillomavirus (HPV). But everybody gets that anyway and it can't be prevented even with consistent condom use. But from an HIV standpoint, unprotected sex with your partner is entirely safe. (And if you want the best protection against HPV, talk to the doctor about getting immunized with Gardasil, the vaccine that protects against the 4 most troublesome types of HPV.)
I hope this helps put things in perspective for you. Best wishes for romance and rewarding sex.