Yes. Lost property belongs to the heirs.
The Shulchan Aruch - Choshen Mishpat - סימן רס - המוצא דבר שמוכח שהנח שם - mentions by the way that lost property belongs to the heirs.
אוֹ שֶׁהָיָה יוֹרֵשׁ שֶׁאָנוּ טוֹעֲנִין לוֹ, שֶׁמָּא שֶׁל אָבִיו הָיָה
(Normally, one has to claim that a found item originally belonged to one. For example, an item is found in an old wall, the owner of the wall cannot claim ownership "because it's my wall". He has to claim it was his and he hid it in the wall; if he didn't do so, we don't suggest the option to him. But regarding heirs, Bet Din would suggest to them that maybe it belonged to their father.)
Regarding selling/keeping the dog - these Halachot are explicit in Shulchan Aruch ibid סימן רסז, in 22, 23 and 24 - how long you have to keep an animal that eats and does not pay for its keep. Somewhere between 3 days and 30 days, depending on the details. After that you can sell it and pay the original owner when you locate him.
How much effort you have to expand on locating the owner is also defined there; seems that announcing it in local shuls is sufficient. Same would apply to locating the heirs.