Yes, I mostly use TBS-T for the primary and TBS for the secondary antibody incubation. In my hands this works for most antibodies. There are however exceptions where you have to use milk or BSA (in TBS-T) together with the first antibody to reduce background.
You can, some antibodies function this way, but it's a good idea to use a carrier when possible. This can be a low concentration, like 0.1% BSA in TBST.
This is possible, and even to be recomended for antibodies which are very sensitive to blocking agents, such as milk. I usually try first with making the antibody up in TBS-T/milk and if this does not work, I switch to TBS-T alone. This may sometimes lead to background being more pronounced. If this happens I normally try to block longer before exposure to primary antibody.