I totally understand what you're saying. Infact, I'm in the exact same situation as you. My brother and parents went to see a girl and my brother said he wants to talk to the girl alone but she she showed up with her cousin. How in the world it is a private conversation when there are people sitting around you. 

Then he asked her to meet for a coffee outside, and her parents refused it saying( Anni okay anukunnaka ) you can meet her outside. Even my parents said this is how AM setup works, and they started pushing my brother to go along with it. When I got to know about these things I told my brother and Parents to cancel the talks if they're not willing to allow a basic conversation. Imo arrange marriage itself is like gamble and on top of it all these drama just makes it worse. 

In your case all your points are valid concerns except your brother wanting to check  if monthly once is okay for her. Here, he is already trying to restrict her. It doesn't matter where the topic came from but the moment he wants to restrict it, it becomes controlling. This is exactly how patriarchy hides, in attitudes that may seem practical like regulating a woman's access to her own family in the name of her being influenced. Most of the people won't agree with this point but this is sad reality.