Yes.
Mosquitos will bite any land vertebrate, though some species specialise, for example biting mainly birds, reptiles or mammals.
Birds can get stung by bees, but do not seem to be so affected by them as we are. My hens avoid hive bees -- this suggests they have each previously been stung (hive bees are not brightly coloured, especially our native European dark bee type).
Some birds eat quite a lot of bees. For example, the aptly named bee-eater will sometimes hawk up and down in front of hives. They prefer the stingless drone (male) bees, but often make mistakes and do get stung by workers -- one bird may have a dozen stings in its mouth at a time. (Drones are a drain on the hive, so the birds are not really harming the bee-keeper.)
Other birds which eat flying insects (such as swifts and swallows) are likely to get stung too. Wasps and bumblebees as well, of course.
I've also seen our cattle get stung by bees as they eat flowery turf. They shake their heads and drool for a bit, then go back to grazing -- so they too must be used to it.