Would be nice if we had more ways to customize them like backgrounds, boarders, letting us actually add the minion to their picture using like the same layering system for our Wardrobe. Even if they can't use clothes backgrounds, effects [stars, bats, etc.] might be a nice way to customize them?

I think the main reason this isn’t done is because there’s a ton of pets and colors, which change their pose. In order to add a wardrobe/pet customization function (like I think neopets has?) they would need to add each piece to fit every pose of every pet.

Your wishes are absolutely possible with the current system though! You can commission an artist here on site to add whatever style, background, effect, etc. you want on your pets and submit them as an overlay!

Also if you didn’t want to go the overlay route (more permanent solution), you can always customize your pets’ profile to make them appear however you’d like (you’d still require an artist’s work though!). But this way you could change how they appear whenever you’d like without having to get multiple overlays done.

Overlays are designed exactly for that.

It's up to you to determine how it can be customized via the overlay system.

No support.

I wouldn't like that, that sounds like how neo has their pets customise system - as mentioned above custom overlays are a much nicer way to customise your pet without it looking tacky (in my opinion anyway..) :)