Q I have been seeing a lot of spinning beachballs lately, and the lights on my NAS seem to be ‘labouring’ at times too. However the NAS seems unable to have its permissions checked, so apart from various log entries, I have drawn a blank. What is causing these problems?

A As your NAS is not directly connected to your Mac, is not a startup volume, and probably is only pretending to run the Mac Extended file system, I am afraid that Disk Utility can do little with it: you will need to use the utilities supplied for it if you think it has problems.

Assuming that the log entries which you supplied coincide with times when the spinning beachball appears, this relates to problems establishing Internet connections. Examples from your log include:
25/06/2015 13:22:51.160 Adobe Acrobat Updater Helper[3474]: Adobe Acrobat Updater encountered errorCode 1001
25/06/2015 13:25:27.304 Mail[2864]: CFNetwork SSLHandshake failed (-9807)
25/06/2015 13:31:20.561 CalendarAgent[280]: tcp_connection_destination_prepare_complete 210 connectx to 98.138.73.131:443@0 failed: Host is down
25/06/2015 13:31:20.897 mapspushd[450]: 2015-06-25 13:31:20.897, 450, 92667d60, [MapsAnnouncements]: server returned error: 404 for URL
and so on.

Failed attempts to connect over a network are a not uncommon cause for spinning beachballs, as most apps will try the connection and wait until that times out before recording the error: during that waiting period, they may well become unresponsive, and result in the beachball showing.

If your Internet connection is still functioning, these suggest that a problem might have occurred in the networking software within OS X. The best way to tackle that may well be to restart: that software will then be reloaded, and the log entries and associated beachballs should disappear.

If they don’t, try a Safe boot, with the Shift key held, and see if that helps. You may be able to work out which software is causing this problem, and update or remove it.