This question brings up some important issues. To begin with, it is clear that conventional home financing in most western countries is based on the idea of mortgaging which, in turn, is based on riba and thus repellent in the extreme to all Muslims. Even so, a number of reputable Muslim scholars have written fatwas which allow, on the basis of legal necessity (darurah), the purchase of homes in the west through mortgages.
These fatwas are based on the assumption, however, that no Shari`ah-compliant alternative is available. Should such alternatives become available, however, the legal necessity will no longer exist, and the fatwas will no longer be valid. Allah willing, in the near future, Shari`ah-compliant alternatives to mortgages will become available to Muslims worldwide.
The first part (a) of the question asks whether or not it is lawful to purchase, for cash, a home financed by the seller through a mortgage which, at the time of sale, has not matured (i.e., the seller has not finished paying off the mortgage and plans to use the cash from the sale to do so). Thus, the concern in this instance would appear to be in regard to hadith prohibiting one's direct and indirect participation in riba. In the case mentioned in the question, a part of the cash payment advanced by the questioner would undoubtedly be used by the seller to pay off a mortgage. In this manner it would appear that the buyer is indirectly involved in the payment of riba.
