Can we? Such as hearts, asterisks, ~, or letters with accents like é? (I really hope so. Because like every pet name I have in mind is taken and I am very much suffering.)

It's not possible :( Space is ok, but any other special character will unfortunately break the name.
Man that sucks. And you know, I vaguely remember a long time ago that using special characters was ok to use. Like I remember seeing a pet with a heart in their name and it was cute, but then I think the site got recoded or something so maybe those pet names are grandfathered in. Oh well. Guess I'll just have to use an alternative.

yeah.. it was possible, but with that "recode" there were a lot of pets in the adoption center with the html code for the specific character because those names were some kind of converted to the written-out code :/
(I suck at explaining and english isn't my mother tongue Dx)