Jesus Christ What the Fuck?!?

I mean I don't know like, 100% for a fact that your landlord can't demand this but, like you said, since your cat is an ESA he would have a hell of a time arguing this is something he's allowed to do.

I had a landlord like this and I'm still fighting her in court for my security deposit.

In regards to contact with him, respond to the requests with something like, 'I'm not going to keep my cat out of the windows or confine her to a cage when I'm not home.' It's polite and mater of fact. I honestly have no idea what he'd reply with.

If he's insistent I'd ask why he thinks this is something you can demand of you.

If he thinks he has legal backing you can ask him to sight the specific law.

Just generally respond in a polite but baffled manner.

Also, save all correspondence. Save emails to a dedicated folder. Screenshot texts. (I assumed I could get my text from my phone carrier but they only save texts for a matter of days in some cases.) If he calls let it go to voice mail.

One way he might respond is by refusing to renew your lease in which case you can either find a non-profit to fight a legal battle over discrimination and retaliation or you can take him to small claims court for moving costs, again, related to the discriminatory (illegal) actions.

Personally giving people these legal head aches makes me feel like I'm standing up for myself and has been good for my mental health.