These are both conditionals: A1 is third conditional, A2 is second conditional. Neither case features a proud father. In case A1 the pride relates to a hypothetical past time, A2 to a hypothetical present or future. From A1: Father would have been proud of you if you had joined the army. From A2: Father would be proud of you (now) if he were alive. Note that as you have written them, A1 and A2 can both be used with either a living or a dead father. A1 can only be used to describe past possibilities which are gone and no longer possible; A2 can describe potential future circumstances which are or are not possible, or imagined but unreal present circumstances.

A1 would be said when referring the the past where as A2 may be referring to the present or future. The second one also would more commonly be said if the father had passed away, like you had previously stated. :-)

The first sentence expresses regret, which refers to something that never happened in reality, and it's exemplified by the past conditional phrase "would have been." For instance, "Your father would have been proud of you if he were alive." This means that because in real life the father is dead, he would never feel proud of his child anymore. In the second sentence, "would" is used in the future tense. This means that there's a possibility the father will be proud of his child one day, and he's not necessarily dead. As for your second question, the first sentence would sound natural and grammatically correct. :)