Some people are quite fussyy particular with using old things, I think I'm one of them 😀
Mattress like you say, definitely. Ideally it should be replaced every 10 years, and everyone has their own preferences on the different type of mattresses. Same goes for sofa, especially fabric ones. I tested so many sofas in the shop when I got mine, you'd be surprised how much 1 inch difference in the height, seat depth or back makes to sitting comfortably!!
As to other stuff, any old stuff from family are not a problem for me, and if they're passed down there is a sentimental value as well. Other old items from 'strangers' such as big pieces of furniture, bookcases, it all depends on the style. Some of them are very nice and good quality. And if you repaint them like you do, it can be very rewarding.
Just curious, is it common to buy a property with all the furniture included in Romania?