"Is here I should stay" should never be used. People will understand what you are trying to say, but the correct wording would be "Should I stay here?" In direct answer to your question, "Is this where I should stay" and "Should I stay here" essentially mean the same and can be interchangeable.

Is this where I should stay? is correct . Should I stay here? is the way to say the second question. If you were planning to visit another city you might look at the hotels close to the beach and ask your friend -is this where I should stay? When you are there you might look at the hotel and if it doesn't look nice ask yourself - should I stay here?

I don't believe "Is here I should stay?" is correct unless you are trying to specify exactly here... perhaps pointing at where "here" represents. I can only think of this sentence being said in a way that illustrates dismay or surprise. For example if someone wanted you to sleep on a dirty flood I would say "Is HERE [where] I should stay?" and point to the place I would be expected to sleep. Also "Is here I should stay?" is not grammatically appropriate so I just would not say it at all. Hope this helps, Nick