Usage – More Than 9 Hundred as Hundreds?

The first example is incorrect. The second and third examples are both correct. Which one you use is mostly a matter of preference, although a hundred appears more frequently than one hundred. There is also another form, an hundred, which was common in the past, but has mostly fallen out of use.

See Google Ngram Viewer: a hundred, an hundred, one hundred:

I'm British, but I can answer for the UK and US:

$100 — a hundred dollars or one hundred dollars?

"A hundred dollars" is how I'd say it in speech. "One hundred dollars" is how I'd write it on a cheque.

$201 — two hundred [and?] one dollar[s?]?

In the UK we'd say "two hundred and one dollars". In the US, they might say "two hundred one dollar[s]".

$1500 — fifteen hundred dollars or one thousand five hundred dollars?

"Fifteen hundred dollars" is how I'd say it in speech. The more "proper" way to say it, and the way I'd write it on a cheque is: "One thousand, five hundred dollars" (never "one thousand and five hundred dollars").

$1525 — fifteen twenty-five dollars or [one/a] thousand five hundred twenty-five dollars?

I'd never say "fifteen twenty-five dollars", I'd either say "Fifteen hundred and twenty five dollars", or "one thousand, five hundred and twenty five dollars". Americans might skip the "and".

For a non-English speaking country, say the number fully using "one", for the sake of clarity. In some countries though (such as the Netherlands and Norway), the use of "fifteen hundred" etc. is the same in that language too.