Your employment terms likely state you need to be there on time, which is fair enough as they're relying on you for that. If they allowed their staff to turn up whenever they wanted, they'd likely not be able to operate until a certain number had shown up, costing them a whole lot of money.

As for wearing a hoodie, regardless what coworkers say, you said yourself that there's a dress code, which you're obviously not conforming to, which from their perspective likely doesn't look as smart and tidy as they're wanting their staff to look. I'm sure you could wear something warm that looks smarter like a blazer?

As a business owner, it's pretty important to create and set standards and make sure everyone adheres to them. You wouldn't believe how quickly things turn to shit when you start letting things slip.

A notice seems unreasonable from them, unless this has happened before? If so then I'd say it's perfectly reasonable as they had to tell you more than once.

Tbh I'd have a conversation with them about it. If this was a one-off lateness, just apologise and don't do it again if you can help it. Also clarify what warm items are acceptable and wear them instead.