Is it possible to get cataracts even if I keep my eyes closed while tanning my face?

Cataracts. There are many causes of cataracts besides uv exposure from a tanning bed-- sunlight, smoking, aging, certain medications, and heredity are all factors. So yes, you can still get cataracts even if you close your eyes during tanning sessions. Also, I am not certain that eyelids will block 100% of the uv radiation from a tanning bed.
Unlikely. Keeping your eyes closed should block all the uv light from the cornea and lens. But why tan with uv? You risk the development of skin cancers (melanomas) and premature aging of the skin (the thinnest on the body). Try one of the alternatives with uv block your eyes will thank you for it.