Does it means that if I have a complete alpha deck or a complete beta deck, both will receive the same XP through gameplay? That clearly favour alpha decks.

While we have not released details on how cards will earn XP through gameplay (we need to ensure that it is not "farmable" by bots) I do expect that alpha and beta cards will gain the same amount of XP from battle. This would make it more valuable to play with alpha cards than beta cards, which I think is a good thing.

Will I be allowed to use cards of both expansions in a deck?

Yes, unless the battle or tournament has a specific requirement for only a certain edition of card you can use some alpha and some beta edition cards on the same team (we are using the term "team" instead of deck since it is not shuffled and you don't draw cards like a traditional card game "deck"). You will still not be able to use more than one of the same type of card. For example, you cannot use an alpha edition frost giant on the same team as a beta edition frost giant.