It sort of does, but the most traditionally grammatical form is "It is I." even in English. Later, English people started to say "me" in this sentence, but that does not change the Czech translation.
The Czech (and original English) structure seems more logical and similar, for example, to German "Das bin ich.".
Yes, it is quite common. Most people first learn this usage as children when they are playing a game. Although, what is most common is saying, "Tag, you're it" to another person. This is not the only context in which "I am it" could be used in English, but more to the point is whether Czech people say "I am it" and apparently they don't.
So, if "Jsem to ja" is how Czech people say the equivalent of "It is me", then "To je ono!" "It is me" is correct and "I am it" is not.
Please excuse my leaving out the Czech accent on "ja" because I don't know how to reproduce it on my keyboard without the little row of Czech accented letters you sometimes provide at the bottom.