Projections don’t have to be limited to the entire earth. You can subset the area represented and still apply a projection. (Though you do have a decision if you’re taking an arc-area to “re-align” your projecting shape so that it best fits your area. But that might be more complicated than you’re looking for, other projections best suited for your locale would be a better fit)
Yes. Why wouldn’t it be?
Projections don’t have to be limited to the entire earth. You can subset the area represented and still apply a projection. (Though you do have a decision if you’re taking an arc-area to “re-align” your projecting shape so that it best fits your area. But that might be more complicated than you’re looking for, other projections best suited for your locale would be a better fit)