Can this Pokemon be considered viable or good for competitive play now?
No, not really, unless that Poochyena happens to have the ideal nature and set of IVs for the set it's using and competitive battling in general. You'll also lose the opportunity to have egg moves, which might cause an issue if there's a move you want that can only be gained through breeding.
Speaking of breeding, it's much more logical/ effective to breed competitive Pokemon rather than catch them in the wild, because then you at least have some amount of control over IVs, and you can basically choose its nature and egg moves as well. Also, this is the reasoning behind my answer to your second question...
Do people do this?
No. Competitive battling Pokemon are all 99.9% bred or hacked in with Pokegen or other software. In-game competitive battling is also pointless in a way, since you can just use Pokemon Showdown which eliminates all the tedious breeding and training involved with making in-game with competitive teams.
Is any of what I said above possible to do?
Yes, besides the part where you catch a level 1 Poochyena. If I recall correctly, there are no level 1 wild encounters in any game save for a few event Pokemon. Level 2 is definitely possible though, and I don't see why an extra level would really change anything.