You certainly can ask the judge to remind the other parent that they haven’t completed the ordered coparenting classes and that you think they will...

An Interspousal Transfer Deed transfers ownership from one person to another, so who did she transfer ownership to?

... and the right not to file for divorce

Yes - it is welfare fraud if she claims the children live with her

I suggest you wait long enough to establish a routine that works and the ask the court to modify the paper to match the actual schedule.

You could try to set aside the order but if you were otherwise served (subserved) and had knowledge of the case, you may not be successful.

If you purchased the house during marriage with money earned during marriage, then spouse is entitled to 1/2 proceeds from the sale.

You have to overcome the presumption against custody created by Family Code 3044 - perhaps by taking a 52-week anger management course, parenting...

What are the papers that you say he never signed?

You have to file in your existing case to request the transfer. It’s a change of venue. Google it.