My DD seemed to be full of loads of mucus for the first few months. It used to wake her up at night in distress, and I think she must've swallowed lots because she would puke up snotty sicks. I don't remember her coughing, but when my kids got to toddler age they would often breath their snot in, and that would result in coughing - thought I'd mention because you said about the mucusy foaming.
Always, always listen to your gut, and keep talking health professionals - could you try a health visitor, because if you get a good one, they can be amazing. They have seen a lot of babies, so will have a feel for if something is not quite right, and they would give more weight when talking to doctors.
Hopefully, this is reassuring, rather than something that will stop you seeking the help your DC needs, because you definitely should pursue it. Everything you say I can think of examples of that from my own two DC. My DD slept constantly, was full of snot, she wanted to be in constant physical contact with someone (even in deep sleep, she somehow knew), and she had a month where she woke up for about an hour in the evening just to scream unconsolably, and then went back to sleep. My DS was crazy alert, needed constant movement, screamed at any toys for not doing what he wanted, was continually fed-up and generally hated being a baby. When tiny, both of them screamed murder at nappy changes, and I really didn't bother with baths because they were a palaver and they both hated it.