Your two symptoms both point to a problem with the power switch. The diagnostics screen is normally accessed (although using it isn't really normal) by pressing the power button and holding it all the way through the TomTom drum sound (20 seconds) and continuing to hold it down for some time until the diagnostic (black and white) screen appears.
A problem with the unit switching itself on and off would also point directly to a problem with the power button. This isn't something that is common to the 1000 unit, else we would unquestionably be hearing it here frequently, so it makes me wonder if there is something about your operating environment that the 1000 isn't designed robustly enough to tolerate.
I see you're in the UK and not somewhere in the deserts of Iraq in the blowing sand, but is there anything at ALL that might be contributing to this in your use model?