I'm sorry Airdroid support but this isn't simply a network instability issue. Running a fiddler session at the time of attempting to log with the windows client makes it abundantly clear that there's a network misconfiguration on your end.
Upon loading the client, it sends the following request: (information in the URL below represented in Caps was replaced to prevent unique identification)
GET
https://srv3.airdroid.com/p14/DeviceAll ... NID_t=T_ID HTTP/1.1
Host: srv3.airdroid.com
Response:
HTTP/1.1 502 Bad Gateway
Content-Type: text/html
Date: Sat, 13 Dec 2014 16:54:37 GMT
Server: nginx
Content-Length: 166
Connection: keep-alive
502 Bad Gateway
502 Bad Gateway
nginx
I have now tried to connect over 100 times. I've reinstalled over 20. I've given this application about 10 times more effort to try and work than any other, ever. Please take another look at your service provider's network configuration because this is what is making this application not work, not "network instability".