.Bach: Whoa, I can't believe some people actually got triggered over this.
Anyway, I love Gog-chan and I hope they keep her
forever.
Darvond: I made a wishlist entry (but it probably isn't live yet) to have her become the permanent mascot of GOG. When I spot it, I'll edit this post accordingly.
I could imagine her holding the forums, posing with game items, and being in awe of game features. Given a bit of refinement or commissioned art and she could be everywhere. Put her in the games, decorate the PDFs with her! Muhahahahah!
Okay, maybe just the former bit.
I like your other ideas. Imagine buying something and the confirmation email has Gog tan saying something like "thanks for your purchase, (username)-kun/chan! I love DRM-free games, but I love *you* even more!"