Take a tin can and put and cover the top with an rubber foil, sealing it air-tight. If the pressure in the tin can is equal to the outside pressure the foil will be flat. If you take it to higher altitudes where the external pressure is lower, it will start to bulge out more and more..
Next glue a stick perpendicular onto the foil. Take a stick with teeth that can rotate a gear when the stick moves in or out. Glue a hand (like on a clock) to the gear. If the pressur in the tin can is equal to the outside pressure the hand is in the zero position. Go to higher altitudes and the hand moves in one direction, go to lower altitudes and it moves in the other. Here's your (very) basic barometric altimeter.
Now, to adjust for changing external pressures also have a hole in the base of the tin foil, into which a cylinder fits very tightly (so it's all still air-tight). Adjust everything so that the cylinder is half in when you're at 0 feet and with an external pressure of 1000 hPa.
Now, if the pressure on the outside is below 1000 hPa move the cylinder out a bit. This reduces the pressure in the tin can and if you move it out by just the right amount the rubber foil again is flat and the altimeter shows 0 feet. If the outside pressure is higher than 1000 hPA move the cylinder in a bit to increase the pressure in the can by the correct amout. Moving this cylinder in and out is what you do when adjust the altimeter for the current local pressure.
Of course, an aviation grade altimeter will be a bit more complicated, but the basic principle is the same.
Concerning your question 2): You never set the altimeter to show 0 feet at an airport that's at an elevation of 4000 feet, you want it to show you the actual elevation of the airfield above sea-level. Thus the pressure you adjust for on your altimeter is the pressure at an elevation of 0 feet, calculated from the pressure at the airport and its elevation, using the Boltzmann barmetric equation. This pressure will be somewhere in the range between 950 and 1050 hPa - or the weather there is so ****ty that you won't want to be anywhere near to it anyway;-)