I create an FTP account in Tools > Accounts, then I try to connect in Explorer.
PhpEd connects successfully, and I can open (download) 1 file from the remote server to edit. But when I try to download any other file, an error occurs:

But I can browse any folders on the server. Additionally, I can browse / donwload / upload files by means of other FTP managers.

Is there some way to resolve this issue? I have full access to FTP server (VDS server).

10054 means that client (phped in this case) received disconnection notification. It may happen if your local firewall decided to disconnect, or your server suddenly disconnected or you have problems with network.
To exclude server problems, try a known to work well public ftp, like for example ftp.mozilla.org
if it does not work too, it's your local firewall or local network-related stuff. If it works, it's something wrong with your server.