Aren't the Little Ones such wonderful helpers. I came home from a scrapbook retreat to find that my son had traces shapes onto one of my cutting mats. No amount of scrubbing removed that ink, but fortunately it was only cosmetic damage.
You've gotten some great suggestions for storage. As to the cutting, lots and lots of practice and patience. And even then, sometimes mistakes happen. I'm sure your cousin will love it. Have you considered putting your cutting mat on your ironing board? I think you'll find it much easier to cut with your cutting surface raised. Or perhaps setting up a space on your kitchen counter after your Little One has gone to bed one night to do a lot of cutting at once.
Cheers, K