For the past three weeks I have started to get frequent communications problems - basically every day on some, but not on all of my computers I notice that after running for some time the client becomes disconnected. Sometimes a messages box appears that says "Manager can not communicate with client", sometimes there are no messages, but in any case it is impossible to exit and restart the manager, and even the "Start" button on the Windows task bar does not work anymore. It feels like the computer is in some sort state of limbo - it is not fully dead yet, but is not fully functioning either. What is most annoying is the only way to bring it back is to do a hard reset.

My Boinc version is 6.10.58

Sounds more like you have some other problem. Usually, the "manager cannot connect to the client" is an effect of something else taking over TCP port 31416, the port at which BOINC does communications between its various parts.

The last thing that apparently does so is Bonjour from Apple, an application for iTunes. Other things are torrent programs (where for each peer a port is opened), virus/worms, other malware.

Check with CTRL + ALT + DELETE -> Open Task Manager what is taking up CPU cycles, what is running. Or let Task Manager run constantly so you don't have to open it on the problem machine.

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Well, I've found out that if I exit the Boinc Manager but let the science applications running everything is OK. It seems like it is the Boinc Manager which is causing the problem. Actually I don't see any reasons to have Boinc Manager running anyway.
When I exited the manager I've noticed that work units are reported when finished and new ones are delivered without having the manager running. It is a bit confusing as I always thought that it is the manager, who is responsible for work units management