[TOMT] [TV] A set of bizarre animated shorts broadcast on British TV around the mid-noughties

When I was pretty young, I came across a series of animated shorts on British terrestrial TV (Either BBC 1, BBC 2, ITV or Channel 4) late at night. I think they were going for a generally comedic tone, but some of them were quite dark. I can't remember the name of the show or exactly when I saw them. To be honest, I'm not entirely sure they were all part of the same show. I do remember some specific plot details of a few of them though.

I'm not sure these plot details are 100% accurate, but they should be pretty close:

One of them was about a couple of sentient sex dolls who wanted to flee their owner. At one point they dreamt about playing on the beach. One of them filled herself up with what she believed was helium, but turned out to be a flammable gas of some sort. Next time she was used, her owner lit a cigarette and the sex doll released the gas at the same time. They exploded.

Another one involved a man wrongfully convicted of a crime (I think it was murder, not sure). Another man was attempting to prove his innocence as part of a TV show. He demonstrated that the convicted man was unable to move quickly enough to commit the crime, as he as physically disabled in some way. The convicted man, however, seemed entirely content with his situation.

Another one involved a man attempting and failing to bond with a boy who I think was his son. The son was rude and standoffish. The man shows the boy music CD, it was one of those 'Now That's What I Call Music' albums, and the boy reacted dismissively. The man ultimately commits suicide by jumping out of a window in his apartment. At the end, the boy reveals that his standoffish persona was a facade.

The only other one I remember was about a friendly seeming old man who other old people were terrified of for no apparent reason. At one point he was introducing himself at an old folks home and everyone starts panicking. One of the old people wets themselves.

If anyone else remembers or knows anything about these shorts please say, I've been intrigued about them for a long time.

Update: I was able to track down one of the shorts (the one about the sex dolls): Blown Away by Hayley Morris. i found info on it on a BBC webpage so it likely would have been broadcast on a BBC channel around 2005, probably along with the rest of the shorts. Unfortunately I don't think the short itself exists anywhere online.