Do animals and birds have souls? Of course they do. I believe they have the same as us. We are basically all the same in that context. In fact, they might be on a higher level than us because they are not cruel and wicked like humans.
Nor are animals usually greedy, egotistical, power hungry or envious. As pets they are loving and devoted. They too can feel sadness and happiness and enjoy play and have fun. They are undoubtedly intelligent and possess natural senses much stronger than ours, in fact senses we no longer have.
Of course, one must define what a soul is. Generally I think of it as that non-material part of us that is a spiritual personality or essence that may or may not move on to another place after death of the physical body.
I don't see why animals and birds should not follow the same pattern as us. In fact if there was no bird and animal life in heaven it could be a pretty dull place.
There is also the possibility that the birds and animals can eventually be reincarnated into humans and we arrogant humans may reincarnate to become birds and animals.
So! What would you like to come back as? Things are not going too well for animals at present in this world. You wouldn't want to be anything that's edible, I think; and man is so intent on killing wildlife around the planet you probably won't last long anyway. Even protected species are not safe because their parts are worth more on the black market.
If reincarnation is a reality I'd like to think that humans who have been unkind to birds or animals will be reincarnated into something like battery hens or force fed cattle, or maybe creatures in a vivisection laboratory. Or they could be dogs in Korea, whales or dolphins in Japan, caged bears in China or bulls in Spain. There's a wide choice.
In their next lives after that those people would most probably be kind to animals.