I think it depends on what your comfort level is. But what everyone does is probably fairly different. You can contract most STIs via oral sex, but it isn't as easy to do as vaginal or anal sex. So generally the risk overall is lower. Some things, like chlamydia or HIV, are generally prevented by using condoms, but others like herpes or HPV are spread skin to skin (so from the base of the penis not covered by the condom to the skin around the opening of the vagina or the mouth/lips). Personally, in all the years I was dating, I never used a condom for oral sex, but I didn't just give it out totally casually and randomly either. I would say it's probably the norm though not to. Obviously, you have to do do what you're comfortable with.
Personally, yes, I always used to go for STI tests between partners. I wasn't having a lot of random sex, mostly serious dating or long term relationships. But I work in sexual health, so for me, getting checked regularly was always just what I did on principle because I think it's smart for everyone to do and I wanted to walk the walk as I was talking the talk to others.
I would say that most responsible adults assume that you'll use a condom unless otherwise discussed at the start of a sexual relationship (I'm in my late 30s, married now, but I don't think I had anyone ever assume we wouldn't use a condom to start, that may be a generational thing though, if you're post-menopausal or much younger, I think it may not be as much a norm). This may just be the sorts of people I've dated though. But having had a sexual health check is a good way to open up the conversation if you think it's heading that way, to talk about contraception and condom use, etc. Yes, generally I have always expected someone to go and have themselves tested before things got serious enough that we decided to forgo condom use (but maybe not always necessarily if we were using condoms 100% of that time, but that was just what I was comfortable with). Anyone who is put off by that level of openness isn't right for you, so I wouldn't feel weird having that conversation.