Nacho, you are so close... Let's get the height of the column of mercury sorted out first. Can you remember what is above the mercury? A (Torricelli) vacuum. The atmosphere pushes down on the mercury opposite to the vacuum. The greater the pressure, the more the column of mercury is pushed into the vacuum, the higher the reading. As for altimeters, the lower the pressure, the higher the reading. Given a day with no pressure gradient, if you took an altimeter that read zero at sea level from sea level and took it to your airport, it should read your airport's elevation. No change in the sub-scale should be required. But here's a question. Do you use an altimeter to measure pressure? I've seen some places that do. You bang on the table, reset the sub-scale until you see the field elevation and pass the value. If that is the case, the point your friend makes is correct.
As for temperature affecting pressure, it's a yes and no. In hot countries with large diurnal temperature variations and big land masses, the pressure changes (and bounces) because of diurnal heating. As the day warms up and the air in the centre of the land mass expands, causing pressure ripples to spread outwards (I'm thinking of Australia here. The QNH is forecast based on this phenomena). What is certain is the the air density us reduced because of in read in temperature, the pressure may remain constant.
Clear as mud?