What does Ichabod do for a money? Is the police station paying him?
He's living rent-free in Sheriff Corbin's cabin. He's still wearing his old clothes, usually - let's hope no one complains about the smell. Abbie brings him food, presumably paid for out-of-pocket. Halfway through the first season, Captain Irving found out that his story is all true, and might have started pulling strings to help or personally chipping in. Both Jenny and Corbin were sort of survivalists because they were familiar with prophesies about the apocalypse, so they probably both have some food and cash squirreled away. All in all, he's probably living frugally but I wouldn't call it a plot hole.
Ichabod's reference to friends among the Mohawk warriors who fought for the Rebels seems at first like an historical inaccuracy considering most Iroquoian First Nations fought for the Loyalists... until you remember that Ichabod was specifically involved in espionage. Who better to be a spy than a member of a nation that the Brits are convinced is on their side?
Katherine's tombstone saying that she was burned for witchcraft seems like another instance of historical inaccuracy but then you find out that she isn't dead and her grave is holding the Headless Horseman's head instead of her body. The mistake was meant to mark the grave so that anyone who knew what really had happened would know which grave the head was buried in.
It's noted on the main page that the Hessians and the Horseman wear British redcoat uniforms instead of the blue that the actual Hessians wore. This is a legitimate case of Artistic License History for the Hessians, but "Necromancer" provides a valid explanation for the Horseman wearing a British uniform: He wasn't a Hessian at all, but a defector from the Revolution (i.e. a British colonist).
Wordof God from the head costumer designer also says it was a matter of practicality; the color blue was banned from both set and costume design by Len Wiseman, to make the feel of the show darker (blue is a calming color and creates the opposite effect of what mood they wanted). Because the Horseman couldn't wear the blue Hessian's uniform, the costume designers instead put all the Hessians in red British officers uniforms.
In the season 1 finale, the Sin Eater encounters a set of prayer beads that burn his hands, but he insists on touching them to learn from them anyway, even as both Ichabod and Abbie protest that they're obviously cursed and it's too dangerous. They aren't cursed; they're *blessed*. He's actually on the side of evil. And he wanted to find out the secrets himself so he could withold details if necessary.
Under Artistic License History on the main page, there's an entry questioning how Ichabod can rant about the cost of things when he should have no way of knowing whether four dollars is a lot of money or not (since the US dollar was established while he was still asleep.) That's exactly the point: He has no idea whether it's a lot of money or not. His inital rant is based entirely off the percentage of tax that Abby was charged for the doughnut holes, not the actual amount of money. He thinks there should be public outrage because of "a ten percent levy on baked goods" and points out that the Revolutionary War started because of a "less than two percent" tax. He assumes that more money is being charged because the tax percentage is higher, but in actuality, it's an amount that would barely be considered pocket change by modern middle-class standards. Ichabod hasn't fully adjusted to the modern day during this conversation, so he's judging the tax by the only standard he's familiar with: One that's 250 years old.
Crane is incredibly progressive for a man from the 18th century. Of course he is! If he wasn't, he wouldn't be The Chosen One because he couldn't function in the modern world and work with Abbie.
Benjamin Franklin's kite flying experiment. As everyone knows, Franklin never actually flew a kite during a thunderstorm. Or did he? As part of the whole Secret War thing against Moloch, Franklin was actually using the "experiment" in an attempt to destroy the famous key, which was actually a magical artifact. So of course he would have denied ever have done so, as there's no reasonable explanation; under normal circumstances it would get a man killed.