Not to you but to the person you stole it from, yes. The underlying legal principle being that though you had the item, you never had the legal control of its ownership, and thus although you transferred the item the original owner retained legal control of it.

Is this true? My instinct is that the person who stole it is still directly on the hook, but I'm not a lawyer.

Intuitively I'd think it's different for cash and goods. A stolen TV used in a barter transaction show of be returned. A dollar bill probably not. But 100k cash? Grey area? if you accept a dollar you can't reasonably be expected to do due diligence on it. But a large sum maybe.

You might have a case against the shopkeeper if you could prove they have the specific bill that was stolen from you.

Being in receipt of stolen property is a crime even if you are unaware it is stolen.

Instincts is not how laws work. You instincts might tell you what is just, but that's not necessarily what is committed to laws.

America is based on common law. It is a pretty cool concept. Law was built up out of what peoples' customs and shared moral conventions were when a judge or jury made a judgement in trials between people. From English tradition. Most counties not descended from English law have this tradition. Law statues passed by governments overlay this older system.

If pawn broker purchases stolen good does the original owner have a right to recover their stolen goods without buying them back? (yes)

That's not even related; currency follows different rules than goods, at least partially because it's fungible.

Yea, the authorities will confiscate the stolen bill and use it as evidence, and then give it back to the original owner.