Allow me to answer your second question first. In regions where there are no corsola protection laws (such as in Alola, where the mareanie line is a native species, and Johto, which features a dense population of corsola), it’s fine to let your toxapex hunt, although Johto restricts how many pounds of corsola a day you can fish for this purpose to avoid decimating the native corsola population. On the other hand, despite corsola being abundant there, the entire nation of Australia strictly prohibits corsola hunting. In short there, it depends on the region you’re visiting. Remember to check local regulations at the start of your journey to determine specific policies. You can even simply ask a league representative when you register for that region’s league.
As for your first question, so long as he doesn’t mind and you’re careful, it should be fine. Just be sure to avoid his fragile body and his venomous spikes when you wake up in the morning.
Best of luck, and congratulations on your new journey!