a) Is there a Halachic OBLIGATION, on sons to bury their father?
b)If yes, is that the case even if they receive no inheritance-e.g. everything is left to the wife?
There is a halachic obligation on sons to bury their father. However, if the father gives away all his money to others, and leaves nothing for the children, they would not be oglitated to bury their father. This would probably apply even to father who leaves his estate to his wife, yet it remains proper to bury the father, at least in a minimal way.
Sources: Shulchan Aruch (Choshen Mishpat 253:31) writes that if a father gives away his estate, but leaves something for his children, they must bury him. Sema explains that they would use the money he left them. In Yoreh De’ah (348), Shulchan Aruch writes that we collect money from inheritors to bury their father, and Beis Hillel writes that this applies even when the father is poor, and the children have independent means. Based on the above ruling in Shulchan Aruch, however, it appears that if a father who has means, yet gives away his estate to others and leaves nothing for his children, then the children are not obligated to bury him (this is ruled by Pachad Yitzchak, kuf; see also Mishpat Kasuv, kuf). However, there is room to argue that this would not be the case when a father gives away his estate to his wife, out of a desire to look after her after his death, and to distinguish this from a father who gives away his estate to others without consideration for his burial costs. Therefore, although according to the principle halachah, the children might be exempt from burying the father, it remains proper conduct to treat the father as though he was poor, and to bury him with independent means.,