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.
Avoid saying "and" in the middle of a number
Bob the zealot is correct that you should avoid saying "and" in the middle of numbers. It is common for Americans to include "and" or "'n" in the middle of a number, especially after the word "hundred". American grade school math teachers discourage this, because it is unclear whether the student has stated a number, or stated a math problem.
As Bob the zealot suggests, you usually should say "and" between a whole number and a fraction. For example, 1½ = "one and a half"; 1.5 = "one point five"; 1¾ = "one and three quarters"; 1.75 = "one point seven five" or "one point seventy five". Similarly, $ 1.75 = "a dollar seventy five" or "one dollar and seventy five cents". Dollars and cents are discussed in another ELL post.
Big round numbers
There are a few different ways to say numbers like 1,000,000,000,000,000.
This number can be unambiguously expressed as "ten to the fifteenth", or less formally as "one followed by fifteen zeroes". Unfortunately, this is not the usual name for the number.
Americans usually express this number as "a quadrillion"; I am told that British speakers used to call it "a thousand billion" (and some still do).
Americans will correctly understand "a million billion"; I am told that some British speakers will think that you mean 1,000,000,000,000,000,000 instead.
Americans use the sequence: thousand, million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, ….
I am told that the corresponding British sequence was: thousand, million, thousand million, billion, thousand billion, trillion, thousand trillion, quadrillion, thousand quadrillion, quintillion, thousand quintillion, ….
I list several ways Americans express large numbers in this Math Educators.Stack Exchange answer: https://matheducators.stackexchange.com/questions/4448/math-activities-for-gifted-second-and-third-grade-math-circle-students/6097#6097
Sizes.com has a history of big numbers, with citations.
Teens
The original post uses examples of most of the words in typical large numbers: units (like "one" or "two"), tens (like "ten", "twenty", "thirty"), hundreds, and large round numbers. The numbers between eleven and nineteen are a bit weird:
- "eleven" = 11
- "twelve" = 12
- "thirteen", "fourteen", "fifteen", "sixteen", "seventeen", "eighteen", and "nineteen" are 13, 14, 15, 16, 17, 18, and 19 respectively.
Examples
Here is how I say the original poster's examples. As Cort Ammon suggests, there is a pause before each triplet of digits. I have indicated the pauses with commas:
999,999 = "Nine hundred ninety-nine thousand, nine hundred ninety-nine"
123,909,909 = "One hundred twenty-three million, nine hundred nine thousand, nine hundred nine"
Best Answer
I recently mentioned answer to a similar question that in some parts of the English speaking world, people say 'and' after 'hundred':
375: three hundred and seventy five
1846: one thousand eight hundred and forty six
etc.
Even if the 'and' is as small as 'n': three hundred'n seventy five