How do I install flex duct to existing duct including registers
I don't believe 7" or 8" ducts are trunks, and therefore you shouldn't be splitting them. Each vent requires its own branch duct from the trunk.

Trunk lines are almost always rigid rectangular duct of 10x12" or larger. You install a starting collar and connect your flex duct to that. 



If your builder didn't properly facilitate new connections for your basement, you may need to have a tinbender come in and extend the trunk appropriately.