Negation and order of words

Sometimes, I don't understand the use of the negation in German.

The basic rule is: Some adverbs, like "nicht", are always placed in front of the part of the sentence they negate. However, you can't put "nicht" in front of a verb in a main clause, because that has to be in second position (English solves this by always using an auxiliary), so you either have to find a different part where negation doesn't change the meaning too much, or, if this is not possible, as an exception you can put it at the very end. "Kein" is just a (most times obligatory) combination of "nicht + ein".

For example here: "Hans kann die Rechnung nicht bezahlen".

Straightforward: "nicht" in front of "bezahlen", so he can't pay.

I have a name so I thought I could use "kein". [Names have nothing to do with it]

If I write "Hans kann keine Rechnung bezahlen", what does it mean, instead?
If would mean "Hans can pay no bill". You negate "eine Rechnung", meaning "such things don't exist, he can't pay any bill whatsoever".

Another question. Is "Hans kann die Rechnung nicht bezahlen" the same as "Hans kann nicht die Rechnung bezahlen"?

No. In this case, you negate "die Rechnung", i.e. a particular bill. So you emphasize that he can't pay that particular bill you were talking about before, i.e. "It's this bill that Hans cannot pay".

However, if you drop the auxiliary, things get more interesting. You can't write "Hans nicht bezahlt die Rechnung", so you write "Hans bezahlt die Rechnung nicht" (he doesn't pay). If you wrote "Hans bezahlt nicht die Rechnung (sondern etwas anderes)", the meaning changes: It's not the bill Hans pays, it's something else.

Danke schön.

Bitte!