Is there any limit on how deep the Server Owner can access my data?
From what I gather in your post, you are on the development side and they are on the administration side.  I can see why they would like to handle the administration portion of the database(s).  That would most definitely include backup/restore, security, etc.  If they are responsible for the maintenance and administration of the data layer, then that is more than appropriate.

As for their suggestions about data normalization, there is a possibility there suggestions are prudent.  At the same time, that would be a database design/development decision which sounds more like your responsibility.  Maybe they were just friendly suggestions, that's up to you to decide.  But if they were forceful "recommendations" then that would be another story, as you application is likely to adhere to the schema.

As for the restore on a development server, it is very shop-specific.  The overruling argument will be who had jurisdiction on that particular server (or servers).  Again, this is very shop-specific.

What I'm trying to get at is there are too many variables, and as participants of DBA.SE there is really no way we can give you sound advice.  I recommend you take this argument to management in your company that can make a decision one way or another.