No this will not give you any problems; as long you are the owner of the domain; you can ask your provider to change it for you though.
Technical contact. The technical contact manages the name servers of a domain name. The functions of a technical contact include assuring conformance of the configurations of the domain name with the requirements of the domain registry, maintaining the domain zone records, and providing continuous functionality of the name servers (that leads to the accessibility of the domain name).
Register.com does this too, and they are a very reputable and respected registrar.
I think it's the proper way to do it, they are in fact the technical handler of your domain after all.
See the post above. The technical contact should be the contact responsible for dns. On first registration, that may well be register.com, until the dns is hosted elsewhere. At that time, a change is due. Other registrations use the registrant for all contacts.