dont know about lebanon, but in jordan yes u can rent, it costs about 500 (at minimum) to rent a good house for a month
for travelling, in bus its very cheap, ull need 20-40 cents to travel anywhere in Amman (the capital), but in taxi its costs lot more, if u took a taxi from one part of Amman and travel (lets say 15km inside Amman) it will cost u about 4-5 Jordan dinars, about 8-9 US dollars
jost a note: if u r coming as a tourist, then taxi drivers and houses owners will ask u for lot more since u r a tourist, if so, then i advice u to do all what u want through a tourism office
good luck, and welcome to M.E
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