Just like kenny said: There is refers to something which's location we know. (There is a robber in a bank in New York) Not including "there is" is correct but suggests that you don't know where the robber is or that this robber you speak about even exists. (I believe there is a robber in a bank somewhere right now but I wouldn't be able to tell you who he is or where he is)

"There is / there are" used to describes something that is known to exist / exists or is at a specific location (in a on certain location).

We use "there is/are" because... we just do. Why worry about it? This is how we speak. :) Yes, both of your sentences are correct.

Both are correct English, however there is a small difference in the meaning and neither are really complete phrases. A robber is in a bank does not 'place' the robber. Because of this, it doesn't 'feel' like a complete sentence or phrase. It doesn't have a place or an action. 'A robber is in a bank counting money' has an action. 'There is a robber in that bank counting money' has an action and location and feels more like a correct sentence. 'There' means 'in that place'. It is the opposite to 'here' which means 'in this place'. Maybe a formal English linguist would be able to explain the subtle differences better.

Actually, I don't see any difference, just as your attitudes are different. "There is" is one A is one. If you think there is sounds smoothly to describe, just choose there is doesn't mean is incorrect. A, Two, or Three you may use to count them. "There are" I don't know how many but I am sure "there is" is one or uncountable.