Undecided questions
How many characters are in hangul?

These are probably not the correct names for them but I think it will do as a basic way to understand the characters:

There are 14 simple consonants: ㅂ/ㅍ/ㅈ/ㅊ/ㄷ/ㅌ/ㄱ/ㅋ/ㅅ/ㅁ/ㄴ/ㅇ/ㄹ/ㅎ
There are 5 double consonants: ㅃ/ㅉ/ㄸ/ㄲ/ㅆ
There are 6 basic vowels: ㅏ/ㅓ/ㅗ/ㅜ/ㅡ/ㅣ
There are 4 double vowels: ㅑ/ㅕ/ㅛ/ㅠ
There are 11 combined vowels: ㅐ/ㅒ/ㅔ/ㅖ/ㅢ/ㅚ/ㅘ/ㅙ/ㅟ/ㅝ/ㅞ

I hope I haven't left anything out.

Numbers can be written in the Arabic numeral style (0,1,2,3,4, ...) or as a word. However, there are two styles for writing numbers in Korean - Chinese style (한나/1, 둘/2, 셋/3, ...) or pure Korean style (일/1, 이/2, 삼/3, ...). Both are commonly used in Korean but they have different functions.

Barry got almost all of it right.
ㅑ/ㅕ/ㅛ/ㅠ are not double vowels, they are iotized vowels, which means they are really a consonant combined with a vowel (y+vowel), keeping in mind that the linguistic definition of a consonant is where the tongue (or any part of the oral cavity above the larynx) must touch some part of the mouth to pronounce.
ㅐ/ㅒ/ㅔ/ㅖ/ㅢ/ㅚ/ㅘ/ㅙ/ㅟ/ㅝ/ㅞ aren't combined vowels, they are dipthongs

Barry has reversed the Korean-Chinese style (more accurately called Sino-korean) and the pure Korean styles.
일 (一), 이 (二), 삼 (三), 사 (士), 오 (五) is Sino Korean
하나, 둘, 셋, 넷, 다섯 is pure Korean
Koreans also have counting words for days (하루, 이틀), numbers of eggs (줄, 판), number of sheets of laver [seaweed] (퇴, 톳), and probably a lot more too.

There are also 11 consonant clusters ㄳ/ ㄵ/ ㄶ/ ㄺ/ ㄻ/ ㄼ/ ㄽ/ ㄾ/ ㄿ/ ㅀ/ ㅄ