Thank you!
To answer your first question, it depends on the species (some pokémon, such as the slowpoke line, never seem to be affected by anything, including events that would logically induce extreme trauma and stress), but with dog-like pokémon, absolutely.
The most important thing to do when you’re taking care of a rescue—and not simply one with PTSD, but absolutely this as well—is make them feel safe. Provide them with a comfortable home. Give them plenty of food, water, and soft toys. Show them affection, but be slow and speak with a soft, easy tone when you’re around them. Remove anything that might seem threatening, and avoid raising your voice, even if your anger or frustration isn’t even directed at them.
Additionally, establish routines with them. Pokémon with PTSD see routines as safe, as maintaining regular habits gives them a sense of control. They know when they’ll be fed, walked, groomed, and so forth, so it’s easier for them to understand what’s going on at all times and to anticipate what you’ll do next. Thus, always be sure to fill your pokémon’s food and water bowls at the exact same time every day (with kibble high in Omega 3 and other nutrients that can boost serotonin), take them out for walks at a specific time and for a specific duration, and so on and so forth. And on that note, be sure to exercise your pokémon, but don’t engage in battles with them until it’s clear they can handle them. Play-fighting with other pokémon may help you gauge when yours is ready.
General notes out of the way, it’s difficult to say what might be causing your lillipup’s distress unless you’ve followed the above notes to a T. In her case, she may benefit from a calming collar (a special collar treated with pheromones or flower oils to give off a calming scent), herbal supplements, or actual drug therapy, depending on the severity of her episodes and the actual diagnosis Nurse Joy helps you reach. (Yes, you may need to take your lillipup to a pokémon center as well, especially if her episodes are particularly severe.)
Alternatively—or perhaps in addition to the above—observe her carefully. Very rarely does a pokémon do something for no reason, so there may be a source you haven’t noticed yet. Keep your eyes peeled especially for subtle movement or sound around your lillipup whenever she has an episode. Don’t forget to pay attention to what’s going on outside as well, as she may also be alerting you to an intruder.
But really, start by taking her to a pokémon center to determine whether or not she does have PTSD and then work from there (although the above-mentioned tips certainly wouldn’t hurt either).
Best of luck, anonymous!