Do you have any other children, or are you pregnant? If so, maybe it's an attention thing!
Could there be a life event that's been a trigger for his behaviour changes, rather than just the terrible twos?
I have 3 sons. My eldest never had a tantrum at all - thought I knew it all then, that I'd got this parenting lark down, etc. Then came my second, and boy did he have some tantrums. Then my third - you've never heard screaming like it. But none of them threw things or hit people, it was more shouting/crying/screaming and occasional throwing themselves to the ground. I found with my youngest if I got my phone out, filmed him for a minute, then asked him if he wanted to watch the video of himself he would stop, watch the video and then tell me to delete it (later he would say don't film me to which I would say I won't film you if you take a deep breath and tell me what's wrong). I would always delete the films and rarely had to start filming once we got to that stage. I think the key was distracting them with something for a moment that gets them out of the screaming place and gives you a chance to talk to them!
I also found talking quietly to them helped - they have to quieten down to hear what you are saying.
I used to pre-empt the getting dressed tantrums by asking him to help me.... I'd then grab a pair of pants and say "ah yes, my hat" and put them on my head. He would fall about laughing and tell me I am wrong. I'd ask if he was sure because it fit like a hat.... and perhaps he needed to show me how to put them on. So he'd grab them off my head and put them on. Next I'd pick up his trousers and say "ah yes, I know what this is. This is my shirt" and put my arm in one of the trouser legs... and so on. He thought it was hilarious and that mummy was silly - but he was dressed before you knew it!
Choices often help nip tantrums in the bud too like someone said further up.
Interesting someone mentioned ADHD - my youngest two are both suspected to have ADHD, but not my eldest. So maybe there is something in that?
I agree with the other posters about the dogs and I think keeping them completely apart until he learns to control his temper would be better.
In terms of hitting other kids, that should always be punished immediately, even if you just got there. So say you are at soft play and he hits another child, you get him to apologise after telling him it was wrong then tell him it's time to go home. He will ask why - make it clear that soft play (or whatever it is) is only for children who don't hurt other children. If it happens during play dates, stop play dates for a while. As a parent who knows your child hits, make sure you are always watching him and look for his triggers to try to stop him before he does it!