Ceiling Register Box Insulation and Airtightness - Is this correct?

I’m in the process of making all the ceiling punctures into my attic air tight and better insulated with caulk and spray foam to reduce moisture and my energy costs. However, the HVAC ceiling register boxes that are installed in my house don’t look like typical ceiling boxes that I’m seeing online. The install looks a bit questionable too. These definitely don’t look airtight and I can’t even begin to think how to make them so.

My questions are:

1. Are these actual ceiling boxes?
2. Is this what would be considered a "typical" new-work install?
3. What would I need to do to insulate and make air-tight?
4. Would it be better to rip these out and replace the register with a old-work alternative such as the Airtec MV diffuser & grill combination?

Thanks in advance for the help!