How would an anarchist handle speeding?
Speeding is a non-crime and would not be "handled". Speeding itself doesn't hurt anyone.
How fast is too fast?
This depends on the conditions, the vehicle, the driver's state of mind and their reflexes. It certainly isn't the same arbitrarily chosen round number for every driver at every time like current speed limits seem to imply. Individual drivers have more information about the limit for safe driving than the state does.
If there was an accident, would someone be arrested?
Maybe. I have no idea what solutions the market would come up with to limit accidents, but one of them could be that you sign a contract allowing your insurance company to lock you up if you commit vehicular manslaughter. I'm sure people could come up with ways of further disincentivizing bad driving, with or without actually having to throw crash-causers in jail.
Who would do the arresting?
Bob's manhunters, inc.. Under contract from your insurance company.
How would I deal with corruption in those firms?
The difference between a corrupt private firm and a corrupt police force (like we have today) is that if you defend yourself against the private firm, they don't show up with a SWAT van half an hour later and start shooting into your house or vehicle indiscriminantly, with the only consequences for them being unpaid leave. Their ability to make a living providing security services is dependent on people trusting them more than their competitors.
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