Many thanks Sebastian. No icons are appearing unfortunately. I should say I'm doing this from the Host account - not sure if that matters. I'm just working with the default installation, i.e. I haven't added/changed skins, etc.
As I mentioned, from the Edit Page menu, the "Stay in Edit Mode" option is not selectable strangely. When I select "View in Layout Mode" what happens is that what looks like an empty login window pops up for a moment and then disappears, the main window is refreshed, but the main window doesn't appear in layout mode, i.e. it just goes back to how it was.
All other functions for Host and Admin appear to work is expected. Page Management from the Admin menu works, but I just noticed that I can't Add Modules from the Modules menu - again what looks like a blank login window momentarily appears, but nothing happens. Adding a page from the Pages menu works OK. I can add and manage Users too.
The site is a fresh install and a test site, so I could clear it out and do it again. But I'm wondering if there is something I've set in the hosting environment that may be causing this? I have another DNN site that has been working fine for years (on DNN 4.9.5!) on the same host provider (Arvixe) and except for choosing ASP 4 instead of ASP 2, all else is the same for this new test site. I did set RWD permissions to the root folder and all child objects for NETWORK SERVICE as a part of my own investigations, but it didn't make any difference.
I chose ASP 4 for the site, because when I went to the site after unzipping the installation file, it reported an error when I had it set for ASP 2. Not sure it that is part of the problem?