Same problem here on my Galaxy Note 2.
Running CyanogenMod 10.2
Android 4.3.1
Chrome 36.0.1985.143
IE 9 (x32 and x64) won't connect at all (lite or full web.airdroid.com)
Because of issues with cell reception, I have to use lite mode most times, and while pictures, videos and ringtones are accessible, contacts, messages, frequents and most other apps don't seem to communicate correctly. I can send SMS by phone number, but it says it failed (even though the phone shows it as sent) Contacts says 'loading...' for a minute or so before just showing 0 contacts.
Please let me know if you need anything else.
Thanks,
Eric