Questions about an automatic gearbox

So I'm learning to drive. My mothers car is an automatic family wagon thing, and my drive home often involves a climb up a windy steep hillside road. My mother has been telling me that I should take it out of 'drive' mode, and put the car in gear 3 while going up the hill, to relieve stress from the engine. I happen to believe she (and her boyfriend who told her) don't know what they're talking about. So does this actually help at all? Is it actually the engine that's getting stressed? I can see the gearbox, or parts of it getting stressed from shifting gears around, but the engine? Is the automatic likely to stress the engine by leaving it in the wrong gear? Also, today I followed her advice (she was sitting next to me :\), and I left it in third all the way up the road. Is third gear acceptable, considering I sat in it from around 15km/h up to about 60? If third is fine for 15k/mh, where to second and first come in? I also managed to cut off some people and run a red light. I don't do enough city driving, there is less to worry about in the country >.<