There is an hourly (more-or-less) shuttle service from Larnaca Airport to Nicosia which stops at Philoxenia Square for €7. It probably costs the same again for a taxi from there to the Ledra Street pedestrian crossing if you cannot manage it by foot with baggage.
As far as the crossing is concerned you DO NOT get your passport stamped by the TRNC authorities, they will give you a piece of paper instead which they stamp. This is because officially the TRNC is an "illegal regime" which is recognised only by Turkey.
I have only gone from South to North for a few days at a time so do not know the time limit but it is the same piece of paper that you would get at Ercan. There are many IT companies who ship their passengers to/from the North via Larnaca.
I cannot comment on travel arrangements within the North as I have always gone by car. All this is well documented on the internet.
Good luck.