Can Botox Help Increase Facial Skin Thickness?

No, Botox use does not result in changes to skin thickness. The skin does thin naturally as we age, and that is a natural process that is unrelated to Botox use.
Botox can be used to purposely reduce muscle size. For example, patients who have masseter hypertrophy (a non threatening soft enlargement of the jaw muscles near the angle of the lower jaw), regular Botox treatments can result in weakening and atrophy of those muscles, creating a smaller appearance.

Botox blocks the transmissions of signals from the nerve to the muscles under the skin. It does not alter the dermal thickness directly. If one has a high resting activity of the contractile muscles before botox then the thickness of the skin and muscle per unit area might be more because of the corrugation of the muscle. After the botox the muscle doesn't contract so it may spread out to its flatter state making the thickness less. If you measure that thickness it would be back to normal.

To the best of my knowledge, botulinum toxin's effect on skin thickness has not been investigated ar defined to any significant degree. Theoretically ongoing use of the botulinum will result in shrinkage of the sweat glands in the skin, as well as other adnexal units responsive to acetylcholine release, which could result in atrophy and decreased thickness. However, this is a real reach.

Botox works on the muscles under the skin and has no affect on facial skin thickness. When wrinkles disappear from the botox effect, it is not because they have been filled in nor peeled off. It is because muscles aren't moving as much to create these lines.

No, BOTOX has absolutely no effect on thickening the skin. For that chemical peeling is helpful. Oddly, ablative laser resurfacing seems to have the opposite effect. However, laser does increase collagen in the upper dermis. The bad news is that laser can cause an atrophy in the more superficial skin which is why sometimes people who have had laser resurfacing look like the aliens have sucked their face off.