Have you checked firewalls? Sometimes they cause annoying internet issues.

Since it's also a laptop, have you tried turning down the graphics settings? You can adjust them out of the game in Documents\My Games\Final Fantasy XIV - A Realm Reborn\FFXIV.cfg (open with Notepad).

Speaking of which, you can also delete everything in that folder (preferably backing it up first by copy and pasting it somewhere). That resets all of your settings (macros, hotbars, HUD Layout), which may help if one of those files is corrupted.