Good God.
He sounds really alarming OP. Stop seeing him, immediately, block and delete.
He's a fruit loop, yes, and a particularly controlling, unpleasant one. He clearly likes very much to have you on the back foot, to make you feel uncomfortable - everything you've related are classic first steps to a seriously abusive relationship developing - criticising, manipulating by going quiet so you panic about what you've done wrong, making things up so that you never know where you are, headfucking you by treating you very poorly (lifts, umbrellas) but acting all normal so you wonder if it's you. Result: he's very quickly managed to train you into being 'under' him, trying to get his approval, being on the back foot.
If you doubt yourself at all, think through the period thing. That is so mad that even in the thick of being headfucked like this, you KNOW he was lying (and you're right - light period, usual carefulness- any woman would raise an eyebrow at that, there would have been no blood - only a bloke with no clue would assume that it was reasonable that you spent your time in the bathroom hosing down the deluge and may have missed a spot
). So have a think through that. Think about his thought processes there. He thought up a way to try and make you feel uncomfortable, stare you out, get you to apologise, then he could 'forgive' you. Chilling, classic, abusive weirdo headfuck behaviour.
He is nuts, and dangerous.
OP once you've got rid of this weirdo, please think about your boundaries and how this has happened so quickly. Here is my take on how that period exchange would have happened, should have happened with someone with strong boundaries:
Him: is everything okay?
Me: yes.
Him: (just stares at me)
Me: why?
Him: there was a drop of blood on the handle in the toilet. (Stare)
Me: oh I'm sorry.
Him: (just stares)
Me: I'm sorry, I've apologised, it was an accident.
Him: (stares)
Me: Is there something wrong? There really is no need to stare at me like that.
Him: (stares)
Me: Right, if you can't accept a normal apology for a normal accidental thing, then I think it's best I leave. Please stop staring at me so rudely, you are being very weird now.
Him: (stares a bit more) it's okay, give me a cuddle before we go out
Me: 'No, I really don't think so. Please never treat me like that again or this won't be continuing.'
You accepted his poor treatment of you, his weirdness and rudeness and fell over yourself to apologise. It goes to show how quickly stuff like this gets normalised in an abusive relationship and how men like this home in on women whose natural instinct is to keep the peace, smooth things over. It's not bad to be a peacekeeper - but at the same time, those boundaries need to stay in place.
'I'm getting wet here, can you just give me MY umbrella back please if you can't hold it in the centre properly?'
'No I don't have trouble washing my hands, I always make sure the sink is wiped from spills, do you mind showing me where all this water is?'
'I chop onions like your mum? (Laugh) That sounds like a good compliment to me!'
'Um, there really isn't any need to show me how to blow out my own candles in my own flat thanks. I've managed so far without drowning in wax, as you can see.'
Of course, there would only be a few of these exchanges needed before someone with good boundaries simply told a nasty controlling weirdo like this to sling their hook.
Get rid, OP, and have a think about boundaries and what you will accept in a relationship.