You've got enough problems to keep you busy for a month! It looks like this plumbing was done in stages by hacks. Lack of venting, wrong pipe sizes, likely incorrect fittings, and Lord knows what else. "Is anything right about this?" I think that about sums up where you are. Considering the scope of work that needs to be done, and the many things that could be done wrong, it might be wise to hire a plumbing contractor rather than trying to DIY. I realize of course that money is likely a concern, but DIY could result in not passing inspection which would require tearing out and redoing, or having a drain system that is no better than what you currently have.