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The dog thing may be an issue. Search for a local realtor in the area.
You could also try craigslist if you dare.
A basement apartment will be 900-1000 per month, studio/1 bed.
Anything above ground with actual bedrooms will be 1200-1500 and up.
There are decent areas surrounding NHP. Franklin Square, West Hempstead, Garden City, etc...
Elmont is close but sketchy in parts.
There are plenty on a facebook group - pet friendly rentals on long island - or something like that. I'm sure you are already on facebook. Do a search for the group and then post what you are looking for if you don't find anything already posted suiting your needs. I think there's a few groups like that available.
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