Just keep it neat and clean I think is the thing.
Definitely doesn't hurt to air on the side of caution. Though they definitely could try to use that in racist ways I presume hopefully they just are trying to stop people showing up with out of bed hair mess everywhere because it makes the bosses feel like their precious employee rolled out of bed 4 minutes before job start and didn't take the time to "get ready" for their day working for them properly. There's a certain mentality among managers who put on their dress shirt and slacks every day after styling their hair and getting their coffee that there is a right and wrong way to start the work day and guess what their way of getting in the right attire and "mindset" is the right way and encourages productivity so they think. Meanwhile the thinking goes getting up late, not getting in the right mindset with the right dress and grooming is the habit of a lazy person who won't give them the 110% they've been drilled to expect in their propaganda mills.
Same thing with the head-covering rule, sounds like they don't want maskers and have a real "we want to see your face and your smile so we know you're not making faces at us, it's part of getting to know you blah blah blah". So I'm thinking that won't be an issue as long as you don't cover your eyes, mouth, and their way of seeing your expressions while on video with them.
Best of luck comrade, you got this!
I presume showered, not particularly gross or oily looking from excessive not washing, not full of crumbs from your meal last night, that kind of thing. And neat = well groomed so reasonably maintained in some way that isn't just frizzy hair floating all over from bed-head. I presume it tends to more apply to anglo hair types where individual strands of hair can just go all over. I don't think it hurts to wear the cap though as you said.
Ultimately it's subjective based on whoever the person judging is but I don't think it's anything to worry too much about.