Okay, so you're in luck, all four faults that you have are because the mini doesn't have those sensors attached to it, so you can ignore them, they come up on mine as well. The ambient air temperature sensor was fitted to the Rover 75, which uses the same ECU.
As for cutting out, it could be unfortunately a number of things, from intermittent wiring to the electric fuel pump, to wires behind the carburettor which short out due to the carburettor rubbing against the bulkhead. It's worth checking to make sure that the engine ground lead is still making decent contact, as this can cause all sorts of problems, which I found with my car. This also included unusual fuelling, and wild sensor readings as well.
There is also the possibility that the ECU has dry joints inside on the connectors, however this is very rare and they are usually extremely reliable.
The other thing to check is if water has leaked down the back of the screen and has water damaged the immobiliser. You'll find this in the passenger compartment behind the dashboard, is held on with two screws and a couple of connectors. Once removed you can gently pry the immobiliser open and check for corrosion, which can be cleaned off using a contact cleaner and toothbrush.
It's in here, that the relays are that could cause the cutting out of your engine, mine had dry joints on, and once resoldered, fingers crossed, I have not had any re-occurrences of any problems.
Remember, if you decide to grab a secondhand immobiliser, you'll need to match up the ECU and immobiliser, otherwise the car will never start. This can be done at the same place that you had the diagnostics machine plugged in.
Generally, faults like this are usually electrical in nature, in other words a connection that is not being made or being shorted out, usually due to vibration as you drive along. Look for fraid wires where they go through the bodywork, most of these have rubber grommets, but they're not always fitted as well as they could be. 
I can't think of anything else offhand, but I'm sure the minute I hit the post button I'll think of half a dozen other things 
Sorry this is all a bit long winded, but as I found out to my own cost, these electronic minis with intermittent faults can sometimes be a challenge. However, once sorted, they are fantastic cars, and are an absolute delight on the motorway. I had to drive for five hours straight the other day, and got out of the car feeling pretty good, whereas with other minis I have owned in the past, I have needed traction after a long journey
I hope some of this helps you find the fault that you are looking for, the very best of luck.
All the best, Bloke
Edited by bloke, 12 June 2014 - 09:58 PM.
