My mother always told me that I shouldn't put pots on the stove without any content, because otherwise they would get damaged. Is this actually true?
I believe that a pan needs to get very hot for damage to occur - 500c is the number I found for cast iron specifically, and carbon steel would be seasoned up similar temperatures, I don't have numbers for copper or other metals pots can be made of offhand but the point stands - it will take very high temperatures to effect a metal pan to the point where it warps or alters itself, and the pan should be safe at any temperature you would be cooking food at. You should be able to heat the pan to those temperatures before any adverse reactions can begin to occur.
On the other hand, it is very difficult to tell how hot a dry pan actually is - it may be too easy to misjudge the safe heat, to let the pan get hot enough to warp or scorch the seasoning or burn whatever is added or worse, suffer thermal shock when something is finally put in the pan. A pan with water in it won't rise above 100c until the water's gone, and while oil won't keep the temp low like water will, it will change as it heats, and can provide some warning that the pan is getting too hot before it reaches a point where the pan is damaged - even if that warning, at the last, is the oil smoking or scorching. Thicker pans should fare a little better, since they take a bit longer to heat, and can distribute the heat more evenly, and they're also sturdier and might resist warping because of that - but thin pans can get very hot on the contact points, and will bend or warp more easily.
Also, with a dry pan the possibility of thermal shock is higher - in a pan with liquid in it, the water or oil acts as a good medium to transfer heat through, sort of like a buffer for temperature change, so if something cold(er) gets added to a hot pan, the oil or water will transfer heat into it and out of the hot pan and liquid, much more evenly, and stabilize the temperature faster. In a dry pan, the heat will transfer only at the point of contact, and fast, so a dry pan will undergo stress at the point of contact much more. Also, adding the water or oil you need to cook with itself can become a source of thermal shock if the dry pan had gotten too hot on the stove, so adding room temperature liquid might cause the pan damage as the heat flows out suddenly and unevenly, pans have cracked under such stresses.
In the end, I think you can certainly put your pans on the stove without any contents, and have the pan come out undamaged even if had been heated quite a bit - but you have to be careful and skilled not to overheat your pans too much, and that can be tricky to gauge with a dry pan. On the other hand, having water or oil in the pan will keep the temperature low or warn you when it is overheating, so it is just a little bit safer (and so probably good advice).
Assuming your mom was talking about putting them on the stove while the burner was on, yes, she was right.
It is ok to warm them slightly just prior to adding a little oil. Some people say it helps the subsequent fried foods not to stick. Don't know if that is true, but don't leave the room and let the empty pot get too hot, which can happen very quickly.
Overheating can cause thin pots and pans to warp. An enamel coating on a pan can chip off, or even melt onto your heating element. "Teflon" type non-stick coatings can burn off and be damaged releasing toxic gases into the air. Even cast iron can get too hot, and the seasoning can be messed up.
You don't often need to. But with a thick base I tend to light the gas the get the oil out, and get the oil hot before adding food. A small amount of oil alone will do very little to keep the temperature down. It will act more as a warning sign.
If you want to dry fry, cast iron is the way to go. This can be enameled. Some flatbreads for example need to hit a hot dry pan. I use a le creuset enameled cast iron pan for this.