I'm a bit worried about the third pic, which looks as if there is some canker where the shoots come from the main trunk. If that is so, then with that tree you would be best to remove it carefully, roots and all and burn it.
However you decide to prune, be very careful to clean the knife in surgical spirit, using an old toothbrush to scrub with, between each cut. Sounds laborious, but it does help to prevent spread.