Well issues, losing water overnight
Looking at the timing of the question, you may have already solved your issue(s), but in the event that you didn't, here's a few tips.

1) Water systems are meant to be used, even if it involves kids who like long showers. I personally like a nice long, hot shower, so in our situation it would have been a test of self-control. Besides, who would purchase a 2-ton truck and be told that you could only put 300lbs in it? Plus, the previous answer estimated 15 litres a minute for a shower - that's a lot more water than any shower I have used. In the U.S., most shower heads limit the flow to under 2 gallons per minute (gpm), which would equate to roughly less than 8 litres per minute.

2) Could be the pump, but the impact of that would be felt much more rapidly. It would probably start to show as soon as the pressure tank lost it's capacity on about the first shower.

3) The sediment is probably coming from exhausting the capacity of your well. I picture the pump sucking water out of the water table until it starts running dry and then getting sediment from the empty cavity. Overnight the well replenishes itself and fills the pressure tank.

What will solve almost all of your problems is a reserve tank, sometime called a cistern. This is just a big poly tank of roughly 1000 gallons, that acts as your main water source. The existing pump supplies the reserve tank with a capacity switch that turns on when it reaches a low level and a separate surface pump supplies your current pressure tank. This will put a lot less strain on your well, well pump and provide constant water supply to your house. It will also make it easier to filter the sediment.

If you are able to change out a well pump at 280', then you can install a reserve tank and surface pump. It's not a complex task and you can save a lot of money by doing it yourself. I built a small shed around the tank to dress it up a bit, leaving enough room for the surface pump and a shelf of replacement filters. The surface pump should be available at the same location you purchased the other pump.

My neighbor also installed a reserve tank and put in two pressure tanks, as he wanted the pressure to remain more constant while he was using it. In any event, this should solve all of your capacity issues and make filtering a lot easier.