And in case my vehicle's plates are from another state, can I surrender them to the New York State DMV?
No, for that, you have to surrender the plates to its state DMV.
Will I get my driver license suspended if I can't surrender my trailer's plates?
Usually, trailers don't need liability insurance, so the DMV will not suspend your driver's license if you can't surrender the trailer's plates.
If I have a trailer, do I have to surrender its plates?
Only if your trailer doesn't have a valid registration.
Can I give my plates to another person?
No, unless that person is going to surrender your plates for you.
What would be the date used as the surrendered plates date?