Hello guys,
sorry for the delay with the answer.

I don’t think the issue is on the routers side. I would take a look on what the SSH client is sending.
On a Linux SSH client, start with the setting echo $TERM …

Wait a second, I am talking about the remote SSH session initiated by GoodCloud.
There are no any problems with running mc from ssh sessions opened from Linux/Windows using putty / remmina with SSH plugin.

Now, to answer the second question:
a) when the remote session (initiated by goodcloud) is opened I am getting the error message:

b) typing F1-F5 generates 4 letters and nothing for F5 (look at the command line promt):

c) pressing F6 works as expected - move/rename: