I only want to answer the first question that was asked. "How corrupt are the police in Panama?"
To answer that I will tell you of what regularly happens in the Volcan, Chiriqui area.
People are falsely under the belief that the police are there to help in times of crisis or duress. If you call because you are being or have been robbed, you will be asked to travel approximately 20 miles to the town of Concepcion to visit the police office there. You will be required to fill out paperwork after which the Volcan police say they will respond in about a week when the paper request arrives. The end result is the trail will be cold after a week and the result is usually one of no response when the paperwork arrives in Volcan.
If you call the police and insist they come, they usually say they don't have a vehicle to come and help. Volcan police have a Ford F250 pickup and a Ford Fusion car. It is an obvious lie when they say they don't have a vehicle to come and help. What people observe in this town is that the car and truck are more frequently seen at roadblocks, friends houses for hours, taking the family to the grocery store, and going to birthday parties. (Only some examples.) (Police policy expressly prohibits using vehicles and equipment for personal use.)
The neighbor lady on our street called the police last November because other neighbors were drunk, playing loud music, yelling, and urinating on other people's cars. All of this lasted well until daylight. The police responded by asking 'what color her panties were'. And, of course, they never came to handle the problem.
A lady friend of mine was stopped at a roadblock. The cop said her tires were worn out and he would be forced to write her a ticket of $150. She did have another option, however. If she went on a date with him he wouldn't write the ticket. She declined and had to come up with the money. None of the tires were worn to the wear indicators.
My wife's cousin was forcibly taken from the back seat of his friend's car because he didn't have his identification on him as they went to the local grocery store. He was beat up in the jail and released the next morning.
An American living here was robbed, beaten, and had his life threatened. The police wouldn't even question the suspect identified by the neighbor.
The police in this town have an agreement with the local tow truck. If your papers are not in order they will tow your car, especially if you are American. Do not be deceived in believing this is a legitimate agreement. The tow truck owner/driver known as Senor Mojica of Taller Mojica will respond to your crisis by telling you lies to gain your confidence. He will say he doesn't like the cops and what they are doing to the people. He will console and promise that he will be careful with your car and wont charge you storage if you can pay the fine and return the next morning. When you return, you will find he is not there and his wife is insisting that you must pay the maximum and not what you agreed to the day before. Of course, you will try to argue the contrary in your poor Spanish. The result is you will pay way more for the tow truck than you will for the fine. When I inquired in the main office in David City, they said the only reason to tow a car is because, 1) you are drunk and there is no one else to drive the car, and 2) the car has been in an accident and cannot be moved under its own power. By the way, when they take (steal) your vehicle they tell you to take a hike no matter where you are or what time it is. They will drive away and leave you standing there alone on the side of the road.
Oh, the fine. That's right. Because you are driving on your US driver's license and cannot be ticketed (because you are not in the system), the cop will write the ticket in the name of the nearest person who has a Panamanian license. This person is most likely a friend or family member in the car with you, or the person who so kindly is trying to intercede on your behalf who witnessed this act of thievery. Of course, you are obligated to pay. Did you understand that? They are taking your property without cause and applying the fine to a person who wasn't driving or wasn't even there.
Just like in the US, there is a penalty for every infraction. But the penalty is not being towed for every infraction. The correct way is that you be written a ticket and then you go and pay it. The cops here in Volcan must not be aware of the little book of traffic laws that describe every offense and its penalty. You will not find any mention in it of towing the victim's car.
But remember, they don't have a vehicle when there is a REAL CRIME.
These are but a few of the greater experiences of corruption in Volcan. I cannot speak for other places but I do from time to time hear similar stories from the Boquete, David, and Concepcion areas. The police here are very corrupt to say the least, judging by the lies and stealing that is happening. They are not your friends.