I was wondering what the week remarks are on horses and how they're decided if that makes sense. Also how do they pertain to matching?

Remarks are just a horse's notes basically, people record the horse's training bars in the remarks so they can match them properly. Say a mare takes longer to level up in Scope, then it's important to find a stallion who levels up in Scope sooner otherwise the foal won't have a very good rating :)