Depends on the app but basically, yeah. For example, the Ivory for Mastodon app is a Mac/iPhone/iPad application, and the "Mac App Store Preview" shows screenshots for all three devices.

https://apps.apple.com/gb/app/ivory-for-mastodon-by-tapbots/...

No.

For example if I buy OmniGraffle for iPad I don't automatically get the Mac version.

There are options for developers to do this but it's not automatic.

If true then consider them gifts.