Hello W H and ArchitGrover,
I apologize for the delayed response on your inquiry. I understand that you cannot seem to get Facebook chat on your Outlook.com account using web browser and Windows Phone to work.
If I may ask, have you tried optimizing your browser to remove all the temporary files and cached information on your browser? If not yet, kindly visit the link below as there are some instances when browser issues are causing Outlook.com not to run properly.
How to Optimize Internet Browser
Also, have you tried using Outlook.com web messenger without Facebook connection? We just want to exclude the possibility that the connected service is having an issue and not the Outlook.com web messenger itself.
Inform us for any results.
Thank you.