If there were an item in my family's possession or something I've found on my land, would a governing body be able to confiscate it if I don't volunteer it?
And conversely, could a museum decide not to give access to certain objects to historians who want to study them?
As to your main question, I don't know much about international laws or agreements(in general), even as it relates to the cultural sector, so I won't attempt to answer it. I know that there are laws based in individual countries that would make you hand over certain found objects.
For instance, the Treasure Act of 1996 in parts of the UK requires those who find something that might be a treasure horde to report the find. Then, if it is determined that the find qualifies, then the treasure can be taken to a museum which would generally be required to pay fair market value of the goods to the finder/landowner.
A bit complicated, right? That's a very basic premise of just law in parts of the UK too, before going into tackle international agreements (which I'm not going to attempt).
As to your second question, museums not giving access to objects to researchers. In general, one of the main purposes of museum collections is their use in research, so for access to be restricted there would generally have to be a compelling reason.
For instance:
The object is too fragile to be handled. In this case, the object might be on a list to be conserved in the future, but currently cannot be handled without risk of serious damage.
The object contains sensitive or classified information. For instance, I'm not going to let someone look at documents mentioning detailing the locations of all of the archaeological sites around where I work unless they are a recognized archaeologist. There would be potential risk (looting for example) in handing that information out to just anyone.
The collection has yet to be fully accessioned and/or cataloged. In this case, the objects haven't entirely been put into the museum system yet, and should be available after they're properly integrated into the collection. This would be why access is often limited to objects recently donated to a museum for example.
There might be some other examples, but that should give you an idea as to an answer. I want to help researchers, I love when researchers want to come in and look at objects and documents, and I really do want to help to the best of my ability, but there will occasionally be times where it can't happen.
I hope that answered some of your questions!