I think the jury is 6 persons. Who chooses the 6 persons?
The size of the jury and the exact process for selecting the jury varies a bit from state to state and the size of the jury in some states varies depending on the which court the case is tried in. But generally the court summons a random group of the general public from various lists like DMV, voter registration, etc. The court will then pull a group of those potential jurors and begin examining each of them to determine if any of them are ineligible to serve on that case. The judge and the lawyers for each side will pose questions to the jurors to uncover any reasons why the person cannot serve. If the person is ineligible, he/she will be excused for cause. Many courts also allow each side a certain number of peremptory challenges. These allow the attorneys for each side to excuse a potential juror without stating any cause. Attorneys use these to weed out people they think will be particularly unsympathetic to their case. After the court finds 6 (or whatever the jury size is) that have not been excused those 6 are seated as the jury and the rest of the people who didn't make it on the jury panel may return to the waiting room to see if they might get picked for a different jury.
Hopefully a jury won't decide if you are liable or not liable.
In most cases they do determine if the defendant is negligent (assuming a personal injury case). Only if the judge has already decided that in summary judgment (very rare) or the parties have agreed on the issue of liability would the jury not decide it.
If they are only considering the amount of an award where do they begin?
The jury in deciding the amount to award (after determining the defendant was negligent) will consider all the evidence presented that bears on the issue of what damages the plaintiff suffered and decide what damages they are convinced the plaintiff suffered and what to amount of compensation to award for that. The judge will give them instructions on how to do that.
Are there cases where the jury somehow found out the rejected settlement offer and that is what they debate.
Generally information about settlement offers discussed by the parties before trial is not admissble at the trial and the jury will never learn of it.
If a driver's liability policy paid a % of the property damage this implies they were found liable by their own insurance company. Can they appear before the court and try to get themselves to be found not liable for personal injury?
The insurer’s determination of fault is not admissible at trial and the jury will not hear what the insurer paid or offered to pay. Generally an insurance company will not pay anything, however, without getting a release signed by the plaintiff that states the payment settles all claims arising from the accident. That agreement will effectively end any pending litigation on the case.