place to stay: u can rent furnished apartments daily/weekly/monthly for pretty cheap price. here is a link
Sewa Apartemen | Apartment Disewakan | Jakarta Apartments For Rent
there is only one airport in jakarta for international arrivals CGK, west jakarta. if u r looking for a job in small charter companies mostly operating single engines, u better live close to Halim perdanakusuma airport HLP east jakarta
always take Blue bird or Express taxi.
some new regulations might make it harder for u to get a job but its worth trying. good luck