I would think it needs to be 5" deep unless you are going with a smaller depth. Most of the recessed we looked at we're 5" and the opening was a little deeper than 5".

Fiberglass isn't asbestos. You can remove it yourself. Just wear gloves so it doesn't irritate your hands/arms, slice it off above and below where you want the cabinet with a utility knife, and throw it away.

This one is 13.5 inches wide. It has a polished edge instead of a metal banded edge, though. The inside is plastic, with plastic shelves. It has a 17.5 inch high cabinet, so you will have to make your hole smaller. It is also only 3" deep. http://www.cabinetsmedicine.com/product/broan-nutone-b7233-focus-single-door-recessed-medicine-cabinet-2-shelves I would suggest that if you really want to replace the one you have with an actual medicine cabinet instead of using the methods others have suggested, you take it out and feel along the edges to see if you can easily widen your hole. Most cabinets are at least 14" wide. That half inch opens up a whole world of small cabinets to you. If the previous hole is not boxed with wood, you might be able to easily widen and lengthen it to suit a different cabinet.

For size, you can also get customized sizing where cabinets can be ganged together. For example, two 12 inch cabinets ganged together or two 15" ganged together would work where you can add lighting as an option and have interchangeable left or right open doors. It gives you alot of storage over your vanity without looking too wide. And if you want to spend the extra cost, you can have them recessed mounted as well. This will make them look flush with wall and less bulky over any sized vanity. And some of these mirrored medicine cabinets have integrated LED lighting, which can also offer a nightlight with dinner function. So, you will not only have lighting from your main bathroom, but an additional light source. I hope this helps