Oh OP my heart goes out to you. Read about high needs babies, it really helped me understand that no, it wasn't normal but that there was (probably) nothing "wrong" with my baby or the way I was mothering her. For whatever reason, some babies just demand so much more from their parents - more time, more physical contact, more movement, more stimulation. I got NOTHING done when my baby was 3 months old. She was cross and fussy almost all the time unless I laid on a constantly changing programme of entertainment for her, held her constantly, she spent every evening screaming her head off resisting sleep and from the age of 5 months woke up every 45 mins - 1.5 hours all night every night until she was almost a year. I thought I would go raving mad down days.
Do go to the GP and get him checked for reflux, CMPA etc because that can contribute (my baby is 15 months now a d I'm finally getting to the point of confirming she probably is intolerant to dairy in her or my diet, wish I'd figured this out months ago!). But don't listen to the people on here saying the reason he's fussy is because he's overstimulated - no knackered new mum goes out of their way to make life hard for themselves, you're stimulating him because he's crabby if you don't, not the other way around. And please don't worry that the poor sleep will harm how development. My daughter was crawling by 6 months, walking by 10 months, and now talks a blue streak and can count to 5, all well ahead of her milestones - so suck on that all the bitches who told me it was my fault she didn't sleep because I hadn't "trained" her to and that she wouldn't develop properly because she wasn't getting enough deep sleep. To the contrary, I still believe that part of the reason she didn't and still doesn't sleep brilliantly is because she is so clever - her brain and body are so busy I think she struggles to switch off.
Also, to cheer you up, with each skill she mastered I saw a marked improvement in her mood and her ability to self-entertain which made my life easier in increments. One kind La Leche leader said to me when I was at my wit's end that some babies just hate being babies, and won't cheer up until they get to a stage of independence they can live with - definitely proved to be so for my girl who is now a mostly happy, pretty independent if crappy -sleeping toddler who amazes me daily with how very very smart she is. It Will. Get. Better. XXX
Things that helped me (as someone who couldn't sleep Train in the end - she wouldn't have it and I couldn't stand it):
feeding to sleep if it works. I know all the books say not to but if it's the difference between getting up 10 times a night for ten minutes or 6 times a night for any hour, you do what you have to do.
Following on from this, side lying to feed in bed and co-sleeping - you can doze while feeding and you don't waste any sleep opportunities trying to "make the transfer". Give it a try one night when your husband's away and see if it helps you get more rest.
Stop trying to make baby happy all the time - if he's clean, fed and you need to get things done, just bung him in the sling (velcro babies really need a good comfortable sling!), tell him you're sorry he's feeling sad and do whatever it is you have to do (prep dinner, walk dog, whatever it is). It's relentless and soul destroying trying to cheer up a crabby baby who gets bored with whatever you try every 5 minutes. It will make you dislike your baby. Accepting he's just feeling shitty and sympathising with that while not letting it take over everything will allow you practically to function and emotionally connect with your baby rather than feel like you're battling him into contentment (and failing!). He's going through a lot. Let him feel his feelings and be there for him. It will probably make him no crabbier and be a lot easier on your heart and mind!
Seriously though it will be ok. You're doing your best. Think how fast pregnancy went. In hardly any time you'll be looking back on this and everything will be different (and probably much easier). Hang on in there!!!