[desuq]

I've lived in Korea off and on in the past and am living here now. I don't know regarding F-visa types, but I think you'd be able to get a job without too much hassle, maybe just some extra-work to seek out companies willing to communicate in English. There's definitely a pool of companies / startups interested in foreigners / their skillset though of course they would appreciate it if you could communicate a bit in Korean.

I believe you can also get a remote job from a company outside of Korea as others have mentioned the currency and work is not involved with Korea so it should be okay.

Jumping in and out with tourist visa is a bit more difficult now with most countries on lockdown. I also think Korea has a mandatory 10-day Quarantine at home or government facility for anyone entering the country, regardless of their vaccine / visa status.

For the last point, if you enter on a tourist visa (B-1,B-2) if it's applicable to you, that visa can be changed within Korea. But if it's the other type tourist visa (C-1,C-2) you can't change the visa within Korea. This is a rule I recall vaguely, so don't quote me exactly.