Can you download a new app to Apple TV 2nd generation?

The 2nd gen (released in 2010) and 3rd gen (released in 2012) do not run apps, but rather have pre-set 'channels' that are sent via server. Nothing can be added or removed by the user (though icons can be hidden).

The 4th gen (also called Apple TV HD) was the first model to support apps, and runs a completely different system (TVOS).

No, the Apple TV App Store was introduced with the 4th generation.

Older models get their ‘apps’ pre-configured (loaded from Apple’s servers), and users can’t add to that.

Yes, the 4K models work the same as the HD in that regard.