I have to disagree with some of the replies.
Some of the things that your son does is totally normal eg tantrum when he has to leave the park
Some of the things that you're doing is fine eg hug after a tantrum or naughty step. I found a hug was a good way of calming them down quickly and getting them to a listening mode. By listening mode I mean state when you can talk to them about what they did. So if you have to tell him "no throwing cups when angry", waiting until he's calmer after a hug provides a good state where he's ready to listen.
Giving into him because he's sad is not ok. If you've realised that you've been unreasonable and change your mind then back down but you're not doing him any favours. Backing down because he tantrums is subconsciously conditioning him to try that again in the future because he's learned that it works.
Does your h know any 2/3 year olds? If he knows other parents it might be helpful if he spoke to them so he had a realistic idea of what they are typically like.
Age 2 and 3 are the ages where they need to crack the full spectrum of emotions imo. It's annoying and exhausting but it's really important for him to learn how to deal with feelings like jealousy, disappointment, anger etc before he starts school and his peers think 
if he's still at the throwing things in anger stage. At this age they'll be learning the words to express their feelings and learning alternative ways to cope and it'll drive you mad but if he cracks this then he'll be emotionally ready for situations like school. Kids who have parents who regularly give into them will find school stuff when they realise that teachers and other kids aren't going to do the same.
Your son sounds like a good boy who is on the right path but be consistent for his sake. Although you might feel better giving into him, learning that mummy means what she says will make your job easier in the long run as he won't use annoying methods like nagging to try and change your mind.