Humm, my Suzi sometimes will kick out her back legs , when she's excited, sometimes, especially when on the grass. However, skidding out? you mean actually rubbing his legs on the carpet? Suzi also stretches her back legs after waking up , but I think that is still different then what your referring to.
He's only 13 weeks old and has been through so much
my gosh, I hope he can recover from these illnessess. I am very glad you took him right away to the vet. Maybe he will outgrow the behavior and it is not anything serious.