Baby's tail is not on fire when Earl pulls him away but later bursts into flames.
Mel Luster's Mating Dance Academy, according to his advertisement, has been in the same location since 72,000,000 B.C. The series' first season takes place in 60,000,003 B.C. It is established in "Hurling Day" that dinosaur "civilization" is less than one million years old, meaning that Mel's business was established at least 11 million years before the rest of civilization. This is in keeping with similar shady businesses claiming to be far older than they are.
Just before Baby gets thrown across the room, both his legs come off of Earl's shoulders for a moment and he is sort of suspended in the air for a second before it cuts away.
As Baby Sinclair sits on Earl's shoulders, hitting him on the head with a pot, the puppeteer can be seen holding up Baby.