For Avatars while in Beta I have had a few done through the new Inventory, such as the one from using a morphing potion (Magical Kauvara), a transmogrification potion (Mutant JubJub), and reading the Grimoire of Affluence (HT-Rich) to a pet.
As for Random Events, well I am not sure but jellyNeo says "On a different note, with the forced retirement of Classic views for converted pages, one thing to be aware of is that Random Events currently do not work on the redesigned website. They simply have not been implemented yet. Since the forced change is live, we're hoping that TNT will consider porting over Random Events for the new site sooner, rather than later."
I assume this means the proper random events like finding neopoints, your pet becoming depressed, and so on rather than avatars. But, I am not sure about them.