So many questions... Slaves that were manumitted received papers declaring their freedom.

Also people born free often needed papers showing they were free. In 12 Years a Slave, Solomon Northup takes the precaution when leaving New York, to get some free papers to bring down to DC with him in case there were trouble. He says the two men (who kidnapped and sold him into slavery) very particularly emphasized to him that he really should get those papers and bring them with him.

Were these papers carried on the person at all times?

Yes, it was a good idea.

How frequently might a free black be asked for their papers?

Again from 12 Years a Slave, Northup says that slaves walking around alone were frequently asked for their papers or other necessary information about themselves and what they were doing there. Because the reward for capturing an escaped slave was high, poorer people were very particular about asking for papers and examining the papers at length. It was great sport for them, though wealthier people weren't so fastidious. If you were a slave and were given a job to do that would necessitate traveling around without a white person to guard you, then your master or overseer would give you a paper explaining who you were, and your circumstances, which you would show to those who asked for your papers. If you had papers indicating you were traveling on the orders of your master, you could show them at any plantation and receive a meal.

What civil authority granted the papers?

The state.

How common was it for these papers to be counterfeited? It seems like this wouldn't be hard to do, especially if freedom papers varied by region (I don't know that they did but suspect so). Someone in region A might not recognize the papers from region B and assume they're legit.

According to Frederick Douglass, who escaped from slavery in Baltimore, trying to get ahold of these papers was a major concern. He said it was very difficult to forge the papers. He couldn't do it and didn't know anyone who could. Paper is something that we take for granted today. It was a more rarified commodity back then, as was ink. Solomon Northup could read and write, but he says in his memoir that he never had access to paper and writing materials, if I recall correctly. Once, he accidentally looked at a note that was given to him to carry, and this caused a whole interrogation, because she wasn't supposed to know how to read. Other slaves didn't know how to read so forgery would have been impossible, a completely insurmountable obstacle. Slave owners were pretty passionate about not losing any of their slaves to escape, so they made sure the papers weren't forge-able.

So how did Frederick Douglass manage to surmount these obstacles? Well, first of all, he was living in Baltimore, which is 35 to 40 miles to the Pennsylvania border--a free state. Second, being in a city like Baltimore allows for a lot more freedom than being on a farm. He had many friends who were free blacks, and he could walk around freely without engendering too much suspicion. His mistress, a northerner, taught him how to read and write. Baltimore is also a port city with endless numbers of free black sailors visiting and leaving on a daily basis. He became engaged to a free black woman who was a seamstress who had her own money. Plus he occasionally got money from his master. His master rented him out on the docks and then collected ALL of his pay, but occasionally gave him a little. He used his money and his fiancee's money and skills as a seamstress to make him "free" clothes that resembled the clothes worn by free black sailors. Douglass couldn't get ahold of free papers but a free black sailor lent him, at great risk to himself, his own papers showing that he was a free sailor. This was before photographs were the norm but the papers contained detailed descriptions of the holder. Douglass looked nothing like the holder (indeed, Douglass is a really distinctive looking guy), so there was another risk he was taking--if anyone looked closely at the paper, they'd see immediately those were not his papers, and he'd get in trouble and his friend whose papers he was using would be in deep shit. But another advantage Douglass had was that railroad. Huge obstacles could be surmounted once the railroad was there and travel didn't have to be on foot anymore, where you could be spotted and it might take days or weeks to walk to freedom, using the stars for guidance. Nevertheless, the trains were closely guarded. He waited until the last moment to get on his train, so that the conductor at the station wouldn't have time to examine his papers closely when he got on. He passed. Then, once on the train, when the check for free papers came, he showed his papers and he was lucky that the examination was only cursory--he didn't look like a slave, with his new clothing. And then within a few hours, he was in Philadelphia, and there weren't any more checks. He had to quickly mail the papers back to the guy he borrowed them from. And he made his was to New York where his free fiancee met him and they went to Massachusetts right away. So even someone as resourceful, intelligent, well-connected, and educated as Frederick Douglass had trouble getting ahold of papers, and he was helped by a whole lot of circumstances that allowed him to surmount a large number of obstacles.

What prevented someone from simply destroying the papers and carrying the person back into slavery (e.g. Twelve Years a Slave)?

Nothing. Which is why there were a lot of cases of free blacks being abducted and enslaved--especially children, from free border states especially. At the time of Solomon Northup's rescue, he was aided by a law in New York that provided money and assistance to rescue free New York citizens who were kidnapped into slavery. The reason they had this law was because this kind of thing happened with some frequency. It's hard to say how frequently, because it was so difficult to escape. It was more common for people to be taken away and never heard from again. It took Northup 12 years to get away, and that was even after him being able to get word to his family almost immediately of what had happened to him. They couldn't find him, and he had no way of getting word to them. He couldn't even mail a letter, and found no one to trust until he met the Canadian guy who helped him. It still took something like 6 months to get away after that, and that was with the aid of several distinguished and well-connected white men, including a lawyer.