First, you should do some research on your particular state. States vary widely on how they settle custody issues. So no one on YA! will be able to answer your questions completely without knowing the state. You can call the court that you are going to and ask them how their procedures work. Which is always a good option. As for me, I have battled in court twice for custody, once in Pennsylvania and once in Montana. Two totally different approaches on how they handled them. In Montana nothing was done without a judge sitting right in front of you and you were able to have witnesses, they came into the court room one at a time, gave their testimony and left right after, and they were kept separate during the hearing so they could not converse with each other.
Pennsylvania did not allow any witnesses. I could not even use my ex-wife's recent arrest against her even though she was convicted and served time and was on probation/parole for selling drugs out of the house my kids were living in. The judge came to us and told us to sit down and each of us had to write out what we wanted. And she warned us ahead of time, that if either of us were unreasonable in what we asked for the other would win and if we were both unreasonable we would both lose and the children would go to foster care. I was not allowed to even mention my ex-wife's history in the document. When we were done, she looked at both of them, my ex-wife, I reckon, was unreasonable because the judge looked at me and told my that I did a good job, she signed it and exactly what I had put down was our custody order. She didn't even have it typed up. It is still in my handwriting.
From these two examples, I hope you see that the judges can have whatever technique they want. I still go back to what I said before, call the court and ask what type of procedure the judge uses normally. Which they may change from time to time.
I am curious about something though. You say right now that you share custody. And from the way it sounds here you share the time pretty evenly also. That being said, why does welfare think he owes child support? That doesn't make sense to me. Especially for $198 per month. I have heard of rich people pay the other parent some support so the child could have a more balanced life while sharing parents, but I have never heard of it unless it is extremely different and then the amount was much larger.
I heard of one case when I was in the last custody battle that I really liked. I don't remember where it was but the judge ordered the children to stay at their home, the one they had always lived in. There were 4 or 5 children in this case also. But whichever parent had custody that week had to bear the burden of moving their clothes in and out every week. I mention this because I think it is the most fair hearing I had ever heard of for the sake of the children. One week dad lived in the house and the next week mom did, they both had to pay equal amounts for electric and other utilities. I also mention this for anyone else who reads this and is going through a custody battle, This may be best for their children.
If I were you, in case the judge will not accept testimony from witnesses or an affidavit from the police. I would take pictures of your home. Lots of them. Show the judge how clean your house is, even behind your fridge, stove, toilet, sinks, and so forth. Take pictures of the food in the fridge. That way your cleanliness is not in question. If the marijuana smell is coming from a hallway that is outside of your apartment I would take pictures of it as well. Explain to the judge that you should have full custody and the father should have liberal visitation. Explain that you have more time for your child. That your ex should pay a little more child support so his son could have a better place to live. That you are doing the best you can.
I would also call and do research on the daycare that your son is in. Most have at least one or two complaints. These complaints you can use to your benefit. It shows you have real concerns about your sons safety, especially when he is with his father. It also would show your ex has not got the best judgment if there are complaints especially if those complaints are listed with the BBB.
I hope this helps some. Good luck in court.