My heart goes out to you OP. My son was very similar to what you have described, and it was awful: the extreme alertness, the long awake times, the relatively "long" stretch of night sleep but with short daytime naps (sometimes only a few minutes, but enough to keep him awake for another 2 hours), the constant vomiting meaning that I was constantly on edge waiting for the next explosion of half-digested milk that had taken so much effort... It was very difficult to bond with him. I felt only a mixture of hatred and pity. And anxiety because he was obviously not getting enough sleep.
It was also extremely frustrating to be given advice by people who did not have a baby like this. I wanted to punch anyone who wrote "put your child down drowsy but awake"! Drowsy? He had two modes: bug eyed alertness or completely zonked out. "A calm mum means a calm baby" was another one that made me want to scream. I would have been perfectly calm if I hadn't been besieged by a highly alert, projectile vomiting, shrieking, nap refuser all day every day, seven days a week, with the added worry of wondering whether he was doing himself developmental harm through lack of sleep.
Anyway, I am so sorry you are going through this. I am not sure whether I missed this being mentioned somewhere in the thread, but something that made things slightly better for me was a bouncer. It might be worth a try at least if you haven’t done so (apologies if you have, and apologies if it sounds like I am suggesting this is a silver bullet -- I don't mean it like that). We had the Baby Bjorn bouncer at home but actually a 15 pound one from Tesco that we bought while travelling was also ok. With the Baby Bjorn one that I mainly used at home, it had the advantage of keeping him upright after a feed (there are several angles you can set it to), and it had a washable cover. I bought an extra cover so there was constantly one in the wash and one on the bouncer waiting to be covered in vomit (muslins on the bouncer are also your friend here). When my son was coming up for the end of his awake time (or when he was zonking out after a feed), I would put him in the bouncer and tap it with my foot. This enabled me to sit and have a drink or look at my phone, or even lie down on the bed at the same time. I used to bounce my son until he fell asleep (fighting sleep every second but about 8 times out of 10 the Baby Bjorn would win) and then after about 15 min of him being asleep I would carefully drop him back into the sleeping position angle it has (and sit with him while he slept because of the worry of it not being as safe as a flat cot mattress). The other brilliant thing about a bouncer is that (at least in our case) when he stirred at the end of his sleep cycle, I could start tapping the bouncer again with my foot and he would often go back to sleep, so that his nap was longer. This also saved my sanity when he went through a sleep regression at 5-6months and I lost even his period of good sleep at night time: keep the bouncer by the cot and put him in it when he stirs, before he has a chance to fully wake. Then try to transfer back to cot when he has drifted off again).
For what it is worth, one day my son started sucking his thumb, when he was about 12 months old, and we went from nap nightmare to nap champion instantly. So being a bad napper now at 4 months does not necessarily mean your baby will always be like that -- there may be hope. In our case, the nap champion period lasted for one blissful year until he turned 2 and then he dropped all naps. Throughout all of this he has remained an excellent night sleeper (except for the sleep regression period I mentioned).
I hope things get better for you soon.