Could having sex cause any cuts or sores without involving an STD?

Yes because you can get all sorts of irritations because of friction not only in membranes which is the case here in what you are talking about now. Moms have rashes sometimes just because they have to hold a baby who is sick and their nappies heat arms and chest or belly when you keep babies near you, it is part of life but you can get healing for some of these things using medications you used before for sore skin and allergies.