The Sole Survivor can interact with the mongrel and will be given the dialogue prompts to ask if there is anything wrong and then a second prompt to render help in the form of a stimpak will be available. Afterwards the right legs will be at full health in V.A.T.S. and the mongrel will walk normally. The third dialogue prompt has either asking her name or choose between the three names Mutt, Sparky or Buttface. If having to ask her name instead of picking one she will then be labeled "Mutt" afterwards. After the naming dialogue concludes Mutt will simply stay in the area until the encounter/zone resets.
She will appear pleased at the Sole Survivor having relieved her pain, sometimes producing a teddy bear and happily tossing it about.
Interestingly, if encountered during Reunions, Dogmeat will be hostile to her. He cannot kill her though and only knocks her down for a few seconds before she stands up again.
Mutt will follow the Sole Survivor around within a relatively large area if they don't walk too far away. The encounter at the Starlight under the Railroad bridge will have Mutt follow as far as the restaurant overhang at Starlight. She cannot be assigned or interacted with in build mode like one would a purchased junkyard dog. Mutt will not interact with a dog house if placed. Pickpocketing a dog collar into Mutt's inventory won't have any effect. Very rarely, Mutt may turn up in random locations after the first encounter if she was aided.