For instance, it’s possible to watch TV and talk on the phone at the same time. You can dance and tell a joke at the same time. But you can’t (I can’t, anyway,) count something and sing a tune at the same time, not without messing one of them up. Why is that?
Actually, you probably can do this. It would be like singing a song but counting the beats in your head at the same time.
As far as the OP is concerned, it’s not that those things aren’t impossible, it’s just that they haven’t been practiced enough. When I started playing guitar, I could go along just fine until I tried to sing a note. I couldn’t understand how everyone else was able to sing and play at the same time.When my playing got better, I could focus more on the singing and the guitar parts would “play themselves”. I think it’s just a matter of practice unless it’s something physically impossible like breathing in and out at the same time or eating your own face.
Well, my friends can tell if I’m watching TV and trying to talk on the phone at the same time. Both activities require your brain to do language processing, so you have to do some time-sharing. So it’s possible to do it, but your performance is degraded. You may not notice if you’re just chatting, though.
But the real answer is that we don’t know in any detail how the brain processes information. But we have some theories that might explain why some activities can take place in parallel while others can’t.
One theory is that the brain acts as a community of agents. Each agent works independently on some low level function and this all gets integrated by higher level agents. The lower level the activity is, the more easily it can be done in parallel. Counting isn’t a particularly low-level activity (it is for computers, but not for humans) so there might not be a part of the brain that’s specialized to do it. Or, it might just take training to do it.
At least one of your examples is doable. In the book “What Do You Care What Other People Think” Richard Feynman has a story about the time he and some friends tried to learn how to judge the passage of time accurately (by counting to 60, trying to hit one minute exactly). One of Feynman’s friends was able to very accurately count off one minute, in his head, while carrying on a running conversation.
The way that the friend’s mind worked is that he pictured a little counter in his head, steadily going from 1 to 60. That was just how he functioned and so it was no problem for him to talk, he just kept a mental eye on the counter and stopped when the number 60 rolled up.
As to why it’s hard…I’ll hazard a guess that if you’re trying to do two activities that use the same part of the brain, the brain has to figure out which one to pay attention to; that gets all the resources and the other process is left running free.
To tie this into the Richard Feynman experiment Valgard brought up (curse you, I came here to say that! 
), when you became more proficient at playing the guitar you were probably adjusting which part of your brain handled the guitar functions, changing it from a aural-speech-interpretive process to a learned-rote-mechanical process.
I have no problem playing piano and singing simultaneously. I can even play the piano and read a book or a newspaper. but I can’t play the piano and talk conversationally, i.e., arhythmically. I know people who can so it must just be a function of where in my brain I’ve stored both of those skills.
One of the few things that people can do well simultaneously is listen to the radio and drive or walk at the same time. So motor skills and language skills don’t seem to interfere with each other.
Most of the time when people say they can do two similar things at the same time (like talk and watch tv) they are rapidly alternating between the two instead of doing them concurrently.
It’s possible to watch TV and talk on the phone at the same time, but it’s much more difficult (completely impossible, for me) to listen to the TV and talk on the phone at the same time. In fact, if I’m trying to talk to someone, I have to mute the TV. Otherwise, my brain can’t figure out which conversation to focus on, and I end up losing my grasp of both of them. I seem to recall that women are better than men at following multiple conversations at once (no cite), so that might have something to do with it.