Can having mulitple ankle sprains cause microtears in Achilles tendons?

Yes. Multiple ankle sprains can lead to micro tears in the achilles and other ligaments around the area. An MRI would indicate if you have torn the achilles and to what extent.
It is possible. Because ankle multiple ankle sprain effect the ankle joint and the muscles around it.
Yes. Mircotears basically means tendonitis. Multiple sprains can cause the achilles to weaken and get tendonpathic. More worrisome is the instability you can have in your ankle without proper treatment. Also with the sprains you can get an ocd (osteochondral lesion) in one of the bones which can cause pain, swelling, locking, giving out, and early arthritis.
Yes. Problems and injuries of the ankle can cause increased strain in the heel cord.