Jaleel.nt,
in the first instance, we deal with a phenomenon that is conventionally known as a passive gerund; the whole clause with being... can be substituted with a noun phrase, eg, ...this award. Therefore, we refer to being... as a nominal -ing participle clause. In the first sentence, it functions as a prepositional complement of as, which requires either a noun phrase or an -ing clause, as we can see here.
In the other example, there is poor word order. The first part (the one before a comma) should be reconstructed as:
His wife being dead, ...
This -ing clause is called absolute - it is not explicitly bound to the main clause syntactically. The normal word order is preserved, however, so the subject must come first. For further information: such constructions are technically known as nominative absolute participial constructions.
Respectfully, Gleb Chebrikoff.