I live in Cyprus so I can offer some suggestions. Try looking at Etihad or Emirates to fly into LCA. You may have to overnight in AUH (that's the route I fly to/from JFK) but I like the schedule. I used to fly through AMS but since Cyprus Airways went bust, I'm not sure what that routing looks like. I wouldn't fly into Ercan if I was going to Limassol - that's just too much driving and hassle if you've never been to cyprus and don't know how the border crossings work.
You are coming to Cyprus during peak season and the hottest and most humid time of the year so prepare yourself. Temps can be in the low 90s.