Allow me to start with the easiest question here: There are quite a few that you could try, including ones for canine pokémon (which absol is considered to be), for dark-types, for pokémon in general … you would really have to be more specific about where in the United States you’d like to go and what sorts of shows you’re considering.
If you mean shows as in contests (that is, you’re considering becoming a coordinator), each region actually has its own circuit, but many of these are much smaller than the ones you would find here in Japan. The two biggest circuits, that of Unova in the northeast corner of the United States and Cielo in the southwest, tend to be difficult to get into and maintain a place in for three reasons. First, it’s extremely competitive, as Americans tend to be rather vicious when placed in competition with each other. Second, there is a preliminary audition for both regions before you enter the circuit proper. This is because of the third reason: contests in the United States are filmed and treated much like any other sort of reality television. For this reason, American organizers tend to be particular about who gets to be on stage in the first place.
Of the two, though, the Unovan contest circuit tends to be easier to get into because there’s a little less competition. While the Cielo contests have no real rivals (aside from the film industry itself), the Unova contests are often overshadowed by pokémon musicals.
On that note, the second easiest question to answer: your absol will likely get along with practically any pokémon. When putting together a team, it’s more important to pay attention to the pokémon as an individual, rather than its species. Occasionally, you may have to handle a pokémon with type biases or known rivalries (obviously, never stick a zangoose with a seviper or a red basculin with a blue), but these are rare occasions. Otherwise, you’d be surprised what can get along with what.
Finally, the third easiest: care.
To start things off, it’s a myth that all dark-types are nocturnal. Absol especially are active in the day or whenever their trainer is. This is because their danger sense compels them to be awake and alert whenever their “charges” are. (Sometimes, that charge is an entire village. Other times, it’s just their trainer.) With that in mind, absol can sleep pretty much anywhere, and they often prefer a hard floor to a bed. You’re welcome to provide one, as well as anything to make your absol more comfortable, but all it will really need is a safe place to sleep indoors.
Grooming is likely the most challenging aspect of owning an absol (other than, obviously, dealing with its precognitive abilities), as it not only has thick fur but also a horn that needs to be well-maintained. Brush your absol daily, and be sure to wash it at least once a month with shampoo designed for dog-like pokémon. (It’s okay to wash your absol between your regular session if it gets into something particularly unpleasant.) Be sure to dry your absol off thoroughly, especially around its mane; it can grow mold if you’re not careful.
When it comes to horn care, your absol will often prefer to maintain its sharpness by grinding it on a rock, so be sure to provide one outside. Brick or a scratching post (normally designed for cat-like pokémon) work just as well. You may also wish to polish your absol’s horn if you’re considering entering it into shows, but be warned that not all absol will allow you to do this. If yours does, use horn polish (generic works just as well) and a soft cloth. Use firm but gentle movements. Don’t pull too roughly, or your absol will protest by bashing you with its horn.
Finally in terms of grooming, be sure to file your absol’s claws every couple of weeks, and clean its teeth daily. This will not only maintain its physical health but also its mental health. Also, some shows may judge your absol based on these traits especially, as claws and fangs are hallmarks of the dark type.
As for extremely basic care, absol are largely carnivorous, just like most canine pokémon. If you aren’t feeding yours specially designed kibble or wet food for its species, make sure that at least half of its diet consists of meat—the leaner, the better. Poultry is ideal, particularly dark meat if you can find enough of it, and eggs are great as your absol grows older. The rest of its diet can consist of a mixture of berries and starchy or colorful vegetables. Avoid fats and fatty foods whenever possible, as well as foods that are typically poisonous to dog-like pokémon (including chocolate, grapes, and so forth). Grains should also be avoided, less because it would kill your absol and more because not a lot of absol can tolerate grains.
In terms of enrichment, an absol doesn’t require much beyond attention, but many enjoy the same types of toys any other dog-like pokémon loves, including balls, ropes, and chew toys. Absol are especially fond of squeak toys, although researchers have yet to figure out why.
As for exercise, all absol should be let out at least once a day (because, well, many are house-trained, so this is how they’ll relieve themselves), so plan for a walk at the minimum. They should also be trained for at least an hour a day, as per usual for any pokémon. Targets are especially valuable when training your absol, as many of its abilities involve its horn, which is, strangely enough, not a thing most absol innately know how to aim with. On that note, always stand upwind from your absol when training it.
Speaking of abilities, the donphan in the room: its “disaster sense.” For the most part, if you treat your absol well, you won’t have to worry about its precognitive ability. Most trainers live their entire lives without their absol’s danger sense triggering because, well, most of us raise our pokémon in fairly safe environments. That said, this is an ability that absol innately know how to use, so very little training is required from it. However, if your absol stops in its tracks for several moments and then bursts into an unusual amount of activity and agitation for seemingly no reason, drop what you’re doing and follow it. That is what its disaster sense looks like, which means that whatever it foresaw, you very likely do not want to be in the vicinity any time thereafter.
It may be worth it to note that if you did want to train your absol to control these visions or to increase its reach into the future, you could pair it with a psychic (either of the human or pokémon variety) for further teaching. Oftentimes, though, it isn’t worth it unless you work with disasters (say, as an EMT, a firefighter, or an insurance salesman).
Best of luck, anonymous!