this dog is stray, or feral, or waif, should be accepted, "al nimănui" (textual, it belongs no nobody) is frequenty used with these meanings (same as ”fără stăpân”, textual, without a master/lord, etc, they all mean homeless, helpless, neglected, abandoned, waif). The word itself comes from ”nimeni” (nobody), from Latin ”neminem” (ne manem, of no man, or more textually, of no hand) which also means abandned, etc.
There is no such rule. You can end a sentence with a possessive. It works like a possessive pronoun, as in the sentence "This dog is mine".
What do you add after the possessive in a sentence like "I'm eating at McDonald's" ?