Without hiring an Attorney, you won't know your rights with respect to any major issues. As to insurance, typically each party will cover their own after divorce. However, additional issues include support, maintenance, and property division. Have you considered hiring an Attorney?
Essentially, the judge may order attorney's fees and reasonable costs in making or resisting an order to transfer the case to another courthouse. This will be based on a number of factors, and the court must first give notice.
When you say "my car", I assume you hold title to the car? If so, have you reviewed the police reports to verify who was at fault? Has the insurance company been willing to cover the damage? And, as you list it as "lemon law" rather than personal injury/property damage, what defect on the car do you believe attributed to the accident? If it was a lemon law, I assume the car was new as well.
With respect to ruining your furniture, if it was mishandled, you could argue that you are owed its market value (under negligence and conversion theory).
Further, you could argue that the moving company committed fraud in attempting to conceal a defect in its work.