Two separate issues here. The problem with connection reset and third party cookies is probably caused by using the
http://web.airdroid.com address, if you use the IP address of the device, the one ending with :8888, it works fine.
The problem with the IP address of the device changing is down to the DHCP server on your router. The DHCP lease defaults to 24 hours on the routers I've used, so if you don't use the device for that time on that router you may get a different address. You may be able to change that to a longer lease.
The support guys here should know all that.