EyeTV3 software works with HDHomeRun - which seems to be about the best bet for DVB these days. Don't take that as a ringing endorsement - I don't use it regularly. (Edit: that sounds a bit negative - I just meant that I don't use it enough to make a recommendation).Can I use my EyeTV 3 software with the network tuners/recorders, with or without Plex? If I need Plex, do to have to buy the paid-subscription level Plex, just to watch and record Freeview or will the free version do?
Yes, I think both Plesk and HDHomeRun's own PVR service need a subscription for recording - you can watch live TV with the free HDHomeRun app or a Kodi plug-in. One free PVR option is TVHeadEnd which can run on a Linux server or some NAS boxes, but its not exactly 'point and click' to set up.
Personally, although off-air TV viewing and recording ought to be a 'killer app' for home networks, its always been a bit of a labour of love and never really gone mainstream, and seems to be dying now, with all the streaming and catch-up options around.