Update.
It seems that changing the remote side router has not solved the problem. There is now much less need to reboot my remote system due to router hanging but issue related to 1k-FA and rotor control remains.
I have three sites where I can control my station from. They are:
Site 1. Windows 7 Professional, Firefox, Avast virus protection, Televell TW-EAV510 router. ADSL line 8M download, 1M upload, 40ms ping with speedtest.net.
Site 2. Windows 7 Enterprise, Firefox, no avast virus protection, very fast fixed line 50M download, 50M upload, 4ms ping with speedstest.net.
Site 3. Windows 7 Professional, Firefox, Avast virus protection, fiber optic line 50M download, 30M upload, 4ms ping with speedtest.net.
All sites use network connection from the same service provider.
Problems I have described never appear when using rotor or 1k-FA web control from Site 2. TCP error counters on RC-1216H's stay on low levels, 0 to just a few.
Site 1 is my primary control site where I experience problems. However, I can easily reproduce same problems in site 3. When checking system info pages of RC-1216H's from time to time, TCP error counters go to 5-20 in just few minutes. When rotor and 1k-FA controls hang very badly they sometimes return to operation without rebooting and in this situation when I read TCP error counter there is sometimes 9-digit reading (hundreds of millions). Reset (removing operating voltage from RC-1216's and connected rotators and 1k-FA) always immediately returns operation. 2 x RC-1216H for rotators and 1 x RC-1216H for 1K-FA uses same power supply. But because I have never problems from Site 2 PSU problem is not very likely. Problems happen totally without RF so shouldn't be an EMI issue.
I have tried removing Avast virus protection but no change in situation.
In worst cases all 3 RC-1216H's can go into a state where they don't respond anymore (this is the situation what I mean above by hanging badly). Just like they would not be even on. In the same time all other IP devices in remote station respond normally, K3 remote works fine etc.
Maybe this is network congestion problem on service provider due to heavy traffic? But why only RC-1216H's suffer so badly from it?