Yep! The gallery for creatu is called the Showroom (you can find it under the Shops tab). Opening a Showroom requires a minimal fee as well as a small fee to upgrade to the amount of space you wish to use.
Do they get frozen at their current age if you put them in there?
Yes, creatu only age when on a profile. Putting them in auctions, rancher shops or showrooms all freeze the pet's age (until they are removed, then they'll continue to age).
Do you no longer have to feed them?
Correct. As they aren't aging they won't grow hungry and won't be at risk of dying. Often users just keep the pets they're most proud of on their profile, and store the rest in their showroom so they don't have to worry about tending to them. If you're going away on holiday or something it's a good idea to shove everything in your showroom until you get back.
Can they be removed from the gallery?
Yep! Once you've got your showroom set up, you'll see an "edit stock" button. It brings up a screen where you can check the boxes beside your creatu and click "remove". When removed they go back to your profile.
Is showing them off the only point of using the gallery, or are there other benefits?
I've basically covered this, but the main benefits of a showroom are that it keeps your profile clutter-free and you don't have to worry about feeding your pets.
Hope this answers your questions. Let me know if you need help with anything else. :)