In short, accurate and inaccurate.
Bounties of the mid 19th C spanned a huge range of rewards from $20-$100 for a single runaway slave (sourced from preserved ads from the 18th and 19th centuries available from the Library of Congress) to $100,000 total for the capture of Serrat, Wilkes-Booth, and Herold following the assassination of Lincoln. The deciding factors were typically availability of funds, and the notoriety of the person with the bounty on their head.
Django, though one of my favorite movies, plays it a little fast and loose with the history of the West in order to reconcile the events of the plot. Django has to save his wife, who is currently a slave, and bring down an entire plantation, all of which has to take place before the passing of the 13th Amendment in 1865 at the close of the Civil War. Before all of the plantation burning and wife saving, however, Django travels across the American West. The issue with this? The time we associate with the "lawless", bounty-hunting, gun-slinging, "wild" west started to pick up around 1865. This isn't to say there wasn't bounty hunting taking place before this time, it has been around since before John Mullowney was hunting down priests in the Irish countryside at the end of the 17th century.
The 1860s and ESPECIALLY the 70s saw a shift in vigilante justice and bounties in the way that they relate to the law. At the time almost all western territories were controlled by US marshals, popularly elected sheriffs and constables, local police forces, and "Indian officers" (John Ball, The United States Marshals of New Mexico and Arizona Territories). Primary accounts of the time collected in Michael Canlis' The Evolution of Law Enforcement in California and WC Holden's Law and Lawlessness in the Texan Frontier indicate a major public distrust with these officials in the western territories. Compound this distrust with the jails that were both few/far between and inadequately monitored, courts in remote regions, and inadequate financial resources to aid in the chase and capture of fugitives and you end up with a population that is ready to take an active role in maintaining order and protecting their individual interests. The reward system we come to associate with the bounty hunter gunslingers of the west emerges at this time because of the physical and financial capabilities of local law enforcement offices (Stuart Traub, Rewards, Bounty Hunters, and Criminal Justice in the West 1865-1900).
Congressional Appropriation Acts initially allocated money for the offering of rewards for the capture of criminals involved in defrauding the government or mail theft. Other acts in the 60s and 70s such as the Postal Service Act of 1867 and the Indian Affairs Act of 1874 included money to pay "bounties" (Traub). The government involvement in the funding of reward monies for criminals allowed the acting commissioner of the Treasury to specifically dictate what constituted "rewardable" action or information and also made it difficult and time consuming for law enforcement officials in remote areas to receive permission to offer rewards for criminals. This delay resulted in the offering of rewards becoming a more local affair instead of waiting for government clearance to do so.
The offered amount of reward money through the US government varied, but we have documentation of AG Brewster authorizing $100 for a mail robber in 1882 (BH Brewster to JL Tidball October 1882 Justice Department Letters Received), US Marshall CP Drake offering $500 for a group of stage robbers in 1878, among many others for those patient enough to scour letters from the Justice Department or newspapers of the time. For further context, the Vernon County Court in Missouri during the 1870s offered up $250 for the capture of any murderer or horse thief (US Work Projects Admin - Historical Records Survey Missouri). These rewards become substantially larger for the capture of more well known criminals and outlaws, and business organizations, understandably upset that their stagecoaches were being robbed and nothing was being done about it, began to offer their own rewards that were much higher than federal or local government rewards (Allan Levett, The Centralization of City Police in the 19th Century). The Missouri, Kansas, and Texas railroads offered $5000 for each member of the Dalton gang (Glenn Shirley, Six Gun and Silver Star). The analysis of various wanted posters from the 1860s to the 1910s (when the Wild West pretty much ended as we know it), indicates that there were a lot of heavy bounties floating around for better known criminals. $10,000 for Boss Tweed, $5,000 for N Appleton Shute, $5,000 for Billy the Kid, $5,000 for Bill Doolin, $25,000 for Jesse James, etc etc.
Considering the notoriety of some of the outlaws of Django Unchained, i.e. the Brittle brothers, it is reasonable for $1,000+ rewards to be offered, but for 1858 considering all of the bigger rewards weren't government funded and privately funded bounties were only starting to be offered in the 1860s.
To answer your second question, "Do we have records of specific criminals and their bounties?" - no. And also yes. It's difficult to track down any typical "record" of bounties. There isn't a leger with a tidy list of criminals and how much the average joe with a rifle and a set of steel balls could make if he brought him in. What we do have is letters, often between US Marshals and the Attorney General granting permission to use federal funds for criminal rewards, WANTED posters, and newspapers. Lots and lots of newspapers. The average person could put out a newspaper ad offering $100 to anyone who can bring their house burglar to justice or return their latest runaway slave. A bank could put out a newspaper ad offering $10,000 for a group of men that robbed them. Newspapers from the 19th century are PACKED with rewards for anyone with information regarding various criminal wrongdoings. Posters and fliers advertising rewards for outlaws are widely available on the internet. So in a sense, yes, we have records.
For further reading I would suggest scouring the Library of Congress Archives, the Nova Scotia archives, papers sourced/linked above, and the Western Historical Quarterly (yes, its a real quarterly). The book Wives, Slaves, and Servant Girls is a great resource for more reward ads from the last quarter of the 18th Century. Other books about the West & law enforcement: The Western Peace Officer, American Law Enforcement: A History, Six Gun and Silver Star, and The Bad Men of the West.
TLDR: $1000+ bounties existed and were offered for notorious criminals, typically by private business organizations, but they weren't common during the time period of Django Unchained. We have records of criminals and how much money was offered for their capture or information leading to their capture.
Happy Hunting.
As an additional question, assuming bounties were issued like this; How would a claimant prove to a magistrate or court official that they had the correct individual in case where they had died?
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