I tend to bruise easier and the bruises tend to stay a lot longer than they have before, though I don't know if that's due to the MS or starting treatment or a mixture of the two. I am guessing a bit of both, as it did start a bit before I started treatment but got a little worse after starting it. I also seem to scar a whole lot more easily, even from something as small as a paper cut, but I am not sure what that really is from.

MS is neurological condition. A bruise is caused by damage to blood cells beneath the skin that collects near the surface of the skin which is not a neurological function. Instead it's the clumsiness/ balance issues as a result of MS that cause one to bump/knock into things that brings about the bruising. Additionally the older you are the thinner your skin and the more easily you bruise with or without having MS.

Yes sometimes I wonder where I've hurt myself....

Yes but I seem to bruise myself more often

MS would not cause you to bruise easier, however clumsiness from MS makes your more prone to accidents. The increased bruising probably comes from age in most cases.