Is "hay" only for present tense?
Are there different words for "there were" and "there will be"?
Yes, haber, used as an impersonal verb, i.e to say there is, there was, there will be, etc has a form for each tense. Have a look here.
Seeing these forms out of context is tricky so I recommend you have a look at some of the lessons we have for "conjugate haber in...". You can search for these lessons in the library. Here are some.
I hope it helps.
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