As Symmetra, can you move your teleporter?
As Symmetra, your ultimate is to place a teleporter.
I don't see any option to move the teleporter and as long as the teleporter is active, you do not build your ultimate's strength.
Is there any way to destroy or move the teleporter?
You cannot move or destroy your own teleporter. However, it does only have 6 charges, so you aren't reliant on enemies destroying your teleporter to reposition it.