Yeah that’s nearly universal. Primary reasons are:

• security: if you leave it to the customer, 99.9% of them will never update router firmware, so nearly all your customers are vulnerable (or already have zombie router)
• lower support costs: if everyone has the same router, much easier for support

This almost certainly varies from country to country, provider to provider. Why not just look up the ones you’re interested in? No blanket answer here is going to be guaranteed accurate.

I don’t think “give away” would be the right concept. More like “included with service”

Isn’t that normal for anywhere in the world?

Yes. And often you can even select the router model out of two or three models they have available.

Yes, although they’re usually pretty cheap and nasty.

I was given a Dasan unit which is pretty grim, but works