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en.wiktionary.org is great for Hangul Super- Newbie learners.
For example, en.wiktionary.org 기계 gives me 3 Kanji (Hanja) meanings/readings for 기계 (gigye) ==> 機械, 器械, 奇計
( btw, in Jp ( Japanese ), 機械, 器械 is Kikai, and 奇計 is Kikei. -- and also the pitch accents (intonations) are different. )
What other Hanja meanings are there for 기계 (gigye) ?
Just answering Q1: Naver's dictionary lists six words.
However, among them, 奇計 is a very rare word. I'm not even sure if I've seen it anywhere, although I might understand the term if I saw it in the right context. The others (氣界/棋界/碁界/器界) are even more rare: I'm sure I've never seen them.
...which leaves us with the two common Hanja spellings, 機械 and 器械. However, whatever dictionary writers may say, I think this should be considered a single word ("기계") with alternate Chinese spellings. Surely the difference in meaning is not large enough to separate them cleanly, and virtually all Koreans would be very surprised to hear that 실험용 기계(machine for experiment) and 동력 기계(powered machine) use different Chinese characters.
In general, I'd advise against trying to look up Chinese characters for every Korean words, at least until you are pretty fluent. Koreans just don't care much about Chinese characters these days, so the word "기계" is to Koreans just that, "기계". Coupled with the sad fact that Korean dictionary writers used to copy Japanese dictionaries without even gathering actual Korean sentence examples, looking for alternate/obscure Chinese spellings can easily give you a false impression of how the word is actually used in modern Korean.
