Is my work for a company that distributes and repairs recycling bins permissible (halal) or forbidden (haram), especially when some bins contain empty alcohol bottles or other waste, and am I required to inspect the bins before repairing them?

There is no harm in your work of distributing, replacing, and repairing recycling bins, even if wine containers are among them, because it is permissible to utilize wine containers for permissible uses, and assisting in such is also permissible. You are not required to inspect the bins when repairing them.