Fresh water pump runs for 4 to 7 seconds

If the pump stops it stays off all is normal and good.
If it comes on with no water use... then you have something to worry about.

Yes it's normal to run a little after the faucet is off. You may also still have some air in the lines or water heater. It can sometimes take a while to get it purged out. Each time you run out of water and pick up air it will act this way and need to be purged again. As you use your rig also pay attention for the pump coming on when you have not opened a faucet. This can be an important indicator of a very small water leak. If you hear the water pump randomly kick on for just a second or so, it's because it has lost pressure. It could be internal pump/valve leakage but it's more likely to be a very slow leak around one of the fittings or hose ends.

The next observation is can a faucet be used for a few seconds or the toilet flushed WITHOUT the pump coming on? Is this a new situation or a new camper? Could it be that an accumulator is in the system without your knowing about it?
We purposely installed an accumulator and it works just like you posted.

Yes this is normal. Once the water flow is stopped at the sink, the pressure needs to be restored in the water lines. Although the water is not really compressible, the plastic water lines will expand a little. Dead ended lines with air trapped in them, like unused laundry faucets or outdoor showers, can also act like an accumulator.

I don't think that's bad. It is just building up the pressure in the lines until a preset point, then it shuts off.

What you need to be concerned about is if the water pump turns on for a few seconds at various times when people are NOT using the water... that means there is a water leak or some sort of pressure leak that should be fixed.

An accumulator tank will keep the water pump from coming on right away, but when it does and finishes, the accumulator tank needs a refill.