You sound like you have given up and are resigned to the fact that your child is out of control due to all the many reasons that "caused it".
Your child is 3, and he is not a medical oddity. He has a bright future and he will respond to tons of love, a routine and firm discipline, like any other child. He is not permanently a child that no sitter can handle, but it sounds like right now you need to fix things.
You are correct. The moving and grandparents and working and not having 2 parents as a united front and having the weak father who doesn't discipline consistently has put things in a tailspin, but you have to stop the downward spiral and be a tough single mom. Stay away from moms with perfect kids who rub your face in it (I don't know any moms with perfect kids).
You are a good mom to realize the sitter with the dog and dangerous home were not good. You have the right idea about discipline and are willing to be firm, but it sounds like you admit it hasn't been consistent especially while butting heads with dad. It almost sounds like you need to move away or stay away from him for a bit, or at least be firm about how you handle your kids. Most moms I know are in charge of child discipline and the dads are just following orders or sort of clueless. Don't let him hold your kids back since he's your ex.
I take care of 3 under 5 usually by myself, and the key is 1)putting their needs first 2) being very firm with discipline BUT 3)Being calm and loving and positive as well as never letting them get away with bad behavior. They can always count on me to be logical, loving, but FIRM, and I'm not afraid to enforce consequences on the second warning, never have been, so now I hardly ever need to.
I would get involved with a church group or Sunday school even if you are not religious. Those parents are usually more strict, and since moving my daughter to a Christian school I see a night and day difference between the firmness of the parents and the behavior of the kids vs the last uppety child psych spoilers at her last school. You need some support from people who are kind who understand you need to climb out of a hole.
If it is at all possible, cut back work to focus on your sons, it sounds like they are too much for sitters right now. I had a friend with 3 kids before I had kids and they were WAY out of control, and since I couldn't discipline them fully since they weren' mine, it really was impossible to watch them.
Regardless, you have to work with the schedule you have and clean up, firm up and make their environment more predictable and positive. Your son probably senses you've given up on him at 3 and knows he is in control of terrorizing everyone. Even if you learn in the future he has some sort of ADHD situation he still needs this structure-even moreso.
You may want to google BILY, (Because I Love You) a parenting support group for troubled kids, focused on reforming parents not blaming kids, they have parent group meetings, a little less strict than Toughlove, and see if they have a chapter near you or some online resources for working moms. Hopefully other moms will have some recommendations for help. Blessings to you.