Do you think that a writer should be able to write on any subject?

Yes, as long as it does not require technical knowledge or specific lingo   

In theory, yes. But the question is then, to what degree. Different people have different styles of writing and match these to a chosen "field", journalists, novelists, even poets, the only thing they have in common, initially, is that they can write. To change "field" you have to change your style and as long as you are prepared to work at it there is no reason why you can't adjust, given as I have said, you have the basics. There are some areas which are more limiting, Technical writing for example where not only do you have to be a writer but you also have to be an "expert" on the subject as this is the one time where creativity and imagination are not  essential requirements.

No. That's a common myth. Specialization makes you more experienced in the field you write in and with connections that make creating stories and finding news much easier if you're a reporter or journalist for example. When you work in the same field, it becomes too simple you know what you're going to ask to get the info and then you write in your own style, a story then varies according to the topic. You might get bored though and it won't hurt to tackle another field then but I advice you not to drop the field you've already established your reputation through. You don't want to miss super opportunities when they knock the door but you should be able to enjoy the topic to write about it more enthusiastically and the reader would feel that.

Write what you know. Or rather, know what you write. That's the rule of thumb.

A writer should be able to write about a topic with enough research, although some items might best be left to those who are professionals in that area. You can write a general article in any area. Some writing will require indepth research. Others simply won't be your genre and no matter how much research is done, it won't come right. Take comedy for instance. Not everyone can write a humorous piece.

Still, don't let that stop you. If you want to write something, experiment with it and see what you come up with.