No. We use "sedang" for continuous tense. So "I'm not living in Japan" is "Saya sedang tidak tinggal di Jepang."
It is correct. Indonesian verbs do not provide tense. The above sentence may be translated as I am not OR I do not OR I have not ... etc. From experience, I have found that most of the time, without providing tense markers (e.g. sudah / senang / etc.) sentences are best translated as present tense. I am yet to hear anyone referring to a past or future event without emphasising these by using tense markers. Having said that, Velonisia's comment provides the preferred way to emphasise present tense