Water-based putty duct sealant is the best choice because it cleans more easily and is maintained for a long time. When using duct sealant with putty on your own, you can apply it with a simple brush. You can also use a caulking gun to apply duct putty, although it's not necessary. In addition, leaks in ducts can cause distant rooms to receive less airflow of the one planned by the duct designer.
Assuming that the cover is sealed to the plaster and that the duct system is relatively airtight, there is no need for an airtight seal between the grid and the cover.