Okay, so I now know what is going on with my pump. I have a leak under the concrete. When we shut the valve on the pipe leading to the house, the pump turns off! (I did check the toilets. Turned the water off to them individually and it still did not stop the pump.) So now I am looking to find someone to A. come and fix it, or B. Rent a jack hammer and try and fix it myself. We shall see!

Not knowing "how" the pipe goes under the concrete and since leaks quite often occur at pipe "connections/joints" if your pipe probably makes a 90 degree turn heading towards the house after entering the ground and then turning 90 degrees to go under the concrete towards your house you might want to check that 90 degree connection/joint if you can easily get to it by a little bit of dirt digging. This 90 turn/joint would be a few inches under the ground before it turns to go under the concrete....unless the pipe goes directly into the concrete and then turns to go under it....don't know what your piping setup looks like exactly where it goes under the concrete.

But for my house even though regular, stiff PVC blue pipe is used to run from the soi water meter to my storage tank/pump and to taps on the fence walls (none of this piping under concrete) when it is time to go to the house/under the concrete slabs surrounding parts of the house the builders used the flexible, black PVC tubing which I guess is less likely to crack/spring a leak with slowly settling soil...it's just more flexible....but once it reaches the house structure/walls it converts back to the typical blue, stiff PVC piping.

Good luck in finding and fixing the leak. Knock on wood I don't develop one under the concrete or in the walls....but it's not uncommon for it to happen...knock on wood.

Well first of all, it is normal to have the pump direct connected to the mains, I think most people have it that way.

Second a non return valve is usually in the pump installed and don't need to be bought extra.

First problem: was very typical the pressure switch.

What you have now could be several things:

1) that red part where the pressure is stored I think in english it is hydropneumatic accumulator (??) is broken (there is a membrane inside) or full with air, or no air behind the membrane.

2) you have a leak somewhere in the tubes/house

3) the non return valve leaks

4) the pressure switch is set wrong. It is like over a certain pressure it turns off the pump. When the pressure falls below a limit it turns it on again. If these two points are too close to each other, it switches all the time (maybe there is just a loose screw for adjusting?).

Anything I forgot?

this fallacy pretty much calls into question every other point you make.