The fact that he doesn't demonstrate these behaviors at school is strong indication that it is NOT a medical/psychological issue.
It sounds like he has a strong personality and is going to require strong discipline at home to deal with it.
Our youngest (10)was like this (and can still be...). It has required years of strong consistent discipline to feel like we aren't hitting our heads against a wall constantly. We never had these issues with the first two, and sometimes I am sick of the necessary vigilance, but it IS getting better. We had to start when he was 3...and this year is the first year I feel like EVERYTHING isn't a battle--in fact, I have been remarkably surprised how few battles we have anymore (the most being when he is tired or hungry...but he is surprisingly even dealing better in those circumstances)
If he argues with one of us, he immediately goes to his room for 15 min. If he argues again (has a tendancy to argue EVERYTHING) he is up there for the night. If there was any verbal lashing out I would do the same thing--against parents, siblings or friends.
I wouldn't worry about his complaining about activities, then being happy after. This is a fairly common behavior. AS LONG as you can get him to the activity, ignore the complaints. BUT if you must physically force him to go (pulling, carrying, etc.)--this is SIMPLE. NO MORE activities.
I don't know how you are responding to his inappropriate rants--your husband says you coddle him. Do you? IF you do, realize that your behavior is encouraging his actions. Tough pill to swallow, I know--I've been there! Now, maybe that isn't it...I cannot begin to know; I can only share my experiences.
But I have taught for 20 years & can say that consistent firm discipline has helped me to have very very few discipline problems over the years.
I was speaking w/ a principal recently who said they have a kindergartener who was crying (RAGING) for SEVEN hours a day. When asked how long she has been doing that she said since she was 2. When asked why, the little girl said cuz if she did it long enough they call her mom & she gets to go be with her sister!!!That principal said, well, game over. You cry, I get you, you stay in my office. period. Behavior was gone in only THREE days! But three LONG days for the principal--but a good investment in the rest of the year in peace.
GOOD LUCK!!!
This!!! The first thing I thought when I read your post was "this lady should meet my 13 yr. old." My just turned 13 yr. old DD act"ed" a lot like what you have described. She is basically over it, but will tell you that she does not like change. She knows she has irrational issues with it, but she is learning to deal with it and thrive.
I'm sure I could have found some peditrician to diagnosis her with something over the years, but I have chosen to stick it out with her issues, ie. change, anger, rage, routines, night terrors, GERD, need to dominate, etc, and now she is almost my perfect child.
My red-headed DD was 3 weeks premature and I was on bedrest with her for months due to an incompetent cervix and premature contractions. She was my first and I was worried and anxious, so I blamed her issues on myself. My husband says she was "born cussin' and smokin' and cigar."

Then at 4 months the colic began until she was almost 7 months. Then I would physically have to restrain her to get her to sleep until we relented and did the Ferber, cry it out method. She also smacked me in the face, intentionally when she was 8 1/2 months old, yes she knew what she was doing, believe it or not. She could physically get herself out of a car seat by 14 months, by contorting her body in such a way to pull her arms out of the neck of her shirts. She also dominated other babies at daycare. Would strip them down and treat them like baby dolls. She would throw violent fits where we would have to restrain her from hurting herself. She would vomit in her bed or wet her pants intentionally (after potty training) to get back at you. She was a peach!
At 2 yrs. old she got a bit better because she could communicate, but not a dream child. But the older she got the better she got. She would hit walls though and have set backs. We had to be very strict with her at times.
Believe it or not, she was a great student for the most part and excelled in school. Then about 3rd grade things changed. She became sick with worry over things and about things that were changing at school, developing GERD and night terrors. We thought we had her issues licked...little did we know what was to come then. In 4th grade I finally took her to a Christian psycologist. After meeting with her the counselor was like "there's nothing wrong with her, she's a great kid." I told her about her anxiety issues and her issues in the past with control, etc. Basically, after meeting with her a few times she told me to back off. My DD couldn't handle all the changes that were happening at school and all the new expectation, etc. So I backed off about grades and she was taught some coping techniques and she gradually improved.
By 6th grade she was on top of the world again. Now she is a straight A honor student, excels in soccer and is very happy socially.
Sometimes I wish someone would have given me a reason for her actions and given her a diagnosis at a young age, but I'm so glad they didn't. She knows her issues and tries to address them herself with my help and we have a wonderful, respectful relationship. She also knows I'm in charge, but it was a long road and I'm sure we are going to veer off course some, but she makes me PROUD!
Hang in there OP. Do the reading and research. See a counselor and maybe he needs to also, but it sounds like to me you have strong willed child, like mine. Good luck....sorry so long!