Yes
The state doesn't repossess your car - private companies do that on behalf of the lienholder. They don't charge you for private property left in your car when they repossess it - that would be illegal. They charge a "storage fee" for the items they remove from your car. Underhanded, yes, but they can legally do it.
yes
They can go onto your property, yes. There are limitations as to what they can do, however. They can't force their way through a locked gate, and they can't enter a garage.
no it can't because the used car is used
As long as there is a lien on the vehicle the lienholder has the right to repossess the property
No, they are allowed to come and take it away.
They've already shown the paperwork - to the police, prior to repossessing the vehicle. They're not obligated to show you paperwork - it's not your car. It belongs to the lien holder.
when you arent making payments for several months when you dont pay your taxes
Yes, if it was not locked. They cannot breach the peace to repossess a vehicle but they can come on your property to get their property, namely the car you do not own. It is their car until you pay for it. So legally they have only recovered their property.
No. A car is personal property.
Yes. But you could potentially sue for any damage caused.
No. To be able to repossess any of your property, they must hold a lien on it. If they have no lien on it, they have no right to repossess. Their only option is to take you to court.