Yes, and perhaps even the vendor, but . . . is this the way you really want to proceed.

Can I settle my case alone?

You may be able to discharge your attorney, but unless waived you will probably still owe an attorneys fee.

What will a counsel usually do if the plaintiff does not accept the offer after the mediation?

Can be either one. Depends upon many factors; none of which have been stated.

When an insurance of the person who caused the accident denies my claim, can I sue the owner of the insurance policy?

Presumably, in TX like most states the owner and operator are presumptively responsible for negligent acts. As a passenger, presumably (again) you...

It depends. His psychological injury may qualify in NC as compensable. Appropriate medical attention and qualified NC counsel should be consulted...

Speak with an attorney. Sooner the better.

Can I reopen a closed car accident case for medical reasons?

Probably not as to the party(s) sued. Speak with the attorney who handled the case.

I don't have enough money to pay for the damage that was caused due to accident?

If sued, your insurance company should remain responsible for the cost of the lawyer but you would remain responsible for anything over the policy...

How much would it be to take this to court

Vehicle and Traffic attorneys are generally fairly reasonable in their fees.

Collision ?

Most properly, as being parked improperly is usually not the case of a crash.