As far as having pets during vet school, I think they're awesome -- I started school with 3 dogs and adopted a 4th early first semester, and they make me happier than just about anything else in the world, help me de-stress, help keep things in perspective, force me to get more exercise than I would otherwise, help keep me warm at night, and I look forward to coming home to them at the end of every day. Sure, it's time-consuming taking care of them, but I love them so it's 100% worth it.
To the OP, have you considered getting your dog certified as a therapy dog? Or getting a canine good citizen certification? That can be a pretty persuasive thing to a landlord -- anyone can claim their dog is well-behaved, but having (at least semi) objective obedience/temperament criteria met, plus references from previous landlords (I second that recommendation! I do it, and I definitely think it helps) carries a lot more weight.
As for the birds and cat, I wouldn't think they'd be too big an issue. Many of the apartments around here allow cats or small dogs; finding someone to rent for 4 big dogs is a little tougher, but definitely doable -- I think being a vet student helps with that, as well.
And for the breed concerns, who could PROVE that he's mastiff/boxer? Most landlords couldn't tell you what mixes my dogs are if their lives depended on it, and I don't have any genetic data showing that they're one thing or another, so I say they're lab mixes. And they very well may have some lab in them, and landlords like lab mixes a lot better than shepherd mix or pit mix or whatever other restricted breed you might think he is. If I were you, I'd say he's a rescue so I can't be sure, but I think he's a great dane mix. You're not lying, since you don't really know WHAT he is, and if he's 70 pounds with a big head, that's not completely outlandish. 🙂