Esp8266 server unreachable. Port doesn't open
The ESP module is not that difficult to work with, if you send the correct parameters to it AND you have the correct power it should work as expected. 

My main suspicions will be cabling (just triple check that connections are as expected) and power source (the ESP is VERY power hungry, if you are powering the ESP from the Arduino that is your problem, you need a very good and stable power for the ESP to work (it might show up on your network but will not have enough power to actually send anything and it will disconnect (that will still show up on your router, it depends on how fast it connects/disconnects but won't be able to send anything).

This link can provide some good information about this subject:

http://bbs.espressif.com/viewtopic.php?t=133