I upgraded to T-Racks 6.
If I delete the T-Racks 5 installation, will I be able to open the existing projects saved in my DAW?
T-RackS 6 is a full new install. Your previous projects will be unaffected. You can choose to migrate them to T-RackS 6 to make sure there are no unexpected changes to your existing projects that are running T-RackS 5.
So no you cannot remove T-RackS 5 until you migrate those projects to T-RackS 6 or render the tracks to audio with the effect printed.