I think it's hard to assess the real problem without a lot more information. Others have suspected it could be an attention problem or jealousy of the baby, which are certainly plausible. I also wonder whether there might be other issues, for example language. How is his speech and communication? My older son went through a phase (thankfully, it only lasted a couple of weeks) of terrible behavior, most notably including pushing other children for no reason, and I was advised that it was probably related to his delayed speech. When he was about 2 1/2, we saw a language explosion, and lo and behold, the bad behavior magically disappeared. It was just his pent-up frustration at not being able to say what he wanted.
I have no idea whether this is the issue with your son, but I just want to encourage you to search for other frustrations he might be experiencing if you don't think this is related to the baby. Even just understanding that my son was frustrated, although it didn't provide any solutions to the behavior problems, helped me to deal mentally and emotionally with all the pushing and tantrums.
Also, has your son always been this way? If so, it could be a personality issue or a result of something that you're inadvertently doing differently with him or -- more likely -- that he's responding to differently than your other children did and do. You also mentioned that he behaves when his dad is home. Why? If it's just the novelty of having dad around, then maybe there's nothing to learn from it, but maybe there is something that is different, either something his dad does or doesn't do or something that you and he together somehow do.
If your son has not always been this way, then maybe the clue to the behavior lies in the onset -- what changed around that time?
One thing I would definitely focus on (another mom mentioned this) is the no napping. I know every kid is different, and maybe some kids don't need naps, but I think the vast majority of 2-year-olds do need naps, or at least a rest of an hour or two. It might be that a little extra sleep will solve a lot of the problems. And, like the other mom said, YOU need the break. I've noticed that when I am frustrated with my older son (now 3 1/2) I am prone to focus on his negative behavior, and my harping only encourages him and makes it worse. If you're exhausted from all the battles and don't get a break, it will be harder for you to focus on whatever positive things your son does. So I would tell him that he must at least take a rest for an hour each day, even if he doesn't want to sleep. I don't know whether your home setup provides this option, but I actually still put my 3-1/2-year-old in a crib for naps, even though he has been sleeping in a toddler bed at night now for about a year and a half. My boys share a room, but at naptime I put the older one in the younger one's crib and put the younger one in a Pack 'n Play in our bedroom. We have a crib tent that prevents my older son from climbing out, and this way he must stay in his crib rather than playing with his toys, and most days he does nap for several hours. Sometimes he elects to just take a little rest, but at least he is actually resting and not playing. Naptime is not negotiable in our house, and it's always at around the same time (1:00). I don't try to decide whether anyone seems tired, and I don't accept any pleas from my kids that they aren't tired (they rarely argue about naps anyway; they know it won't do any good). I work part-time from home, and naptime is a key period for me to get work done, so I really NEED my kids to sleep, and to sleep simultaneously. A lot of my friends wonder how I can possibly get two children to sleep at the same time for 2-4 hours every single afternoon, but I don't think my children are miraculous or that I'm doing anything superhuman. It's just that I need them to nap, so I make it a priority. Try making it a priority -- it sounds like you really need it.