Chemical Peel - How Many Layers of Skin Are Removed?

Peels are typically described in terms of depth rather than layers. Alpha Hydroxy Acid peels, such a glycolic acid, are a light peel which are good for fine wrinkles and uneven pigmentation. Since it is a light peel, the recovery is faster, but you may need multiple treatments. The day after the peel, you are usually presentable to return to your normal activities.
TCA peels are considered a medium-depth peel which are also effective for fine wrinkles and blemishes. The recovery time is longer with a TCA peel versus an AHA peel, but the effect is greater. It generally takes several days for the treated areas to heal.
Phenol peels are deep peels, which are most effective for coarse wrinkles. Caution should be used with having any deep peel as the risk of scarring or permanent damage may result. As for your question about combining AHA and TCA, my recommendation would be to use a daily AHA product in addition to having a TCA peel. It is better to go gently and repeat a peel if necessary rather than peeling too deep and causing permanent damage.

It really depends on how it is applied. Usually it just works on the top part of your skin, the epidermis.

Low strength TCA peels usually remove the epidermis and treat the underlying dermis to a varying extent. Some of this depends upon how it is applied  - how many layers, strength, and the pressure with which it is applied as well as your skin type.