Are eye bags caused by sagging skin and fat bags?

No, freezing fat technique should never be used for under eye fat bags as they can have very bad consequences. The ideal treatment is lower blepharoplasty using transconjunctival technique with repositioning of the fat bags. 

Eye bags are most commonly the result of the natural aging process or simple genetics. There are many things you can do to treat bags, hollows, and dark circles, but a combination approach will give the most improvement. Non-invasive, less-downtime options such as fillers (Belotero, Juvederm Volbella), lasers, microneedling/PRP, Cosmelan peels and stem cells can all help give some improvement. A surgical option to remove skin and/or fat might be the best solution long-term. 
For puffiness and more significant sagging under the eyes, we often use Renuvion helium plasma radiofrequency energy to achieve incredible results and take years off of your appearance. Combining RF energy and cool helium plasma, the signs of aging (wrinkles, pigmentation, skin laxity) are improved in a Nonsurgical Eyelid Lift. The focused heating tightens and peels the upper layers of skin with extreme precision, leaving the surrounding area undamaged, smoothing and tightening the periorbital skin, and improving the overall skin quality around the eyes. It's an aggressive treatment, and the recovery time is longer than with other nonsurgical procedures, but the results can be life-changing.
I suggest getting a formal evaluation with a cosmetic dermatologist to see what is best for you and your goals; our practice is located right here in Los Angeles. Sometimes surgery is best and is always an option. Best, Dr. Emer.

Fat freezing techniques are not designed for eye bags.  Lower blepharoplasty surgery is an option and a more permanent solution.  The most common temporary treatment is fillers.  This option is performed in the office and the results last up to one year.  Both options can be combined with lasers, chemical peels and mcroneedling to improve overall skin tone and texture. 

This is a good question. Cryolipolysis (Coolsculpting) can freeze the fat around your body and make it shrink, including in some cases around the face. The fat around the eyes is not a good area to have this treatment because the freezing technique is too close to the eyes and, even if it was safe, it would be difficult to calibrate the amount of fat you are trying to make go away. Your best bet would be to get an opinion from a qualified oculofacial plastic surgeon who is part of ASOPRS (American Society of Ophthalmic Plastic and Reconstructive Surgery) who focuses only on eyelid surgery to determine what treatment would help the best, be it laser, filler, or surgery. There are many great specialists in your area.

Tear troughs, depressions across the lower orbit with bulging fat pushing out the skin forming what many call "eye bags" are caused by loss of volume in the cheek and stretch of the skin of the eyelid and cheek. as the skin descends, it is stopped by ligaments from the orbital rim to the skin. The bulging is caused because the skin of the eyelid cannot descend any more and the normal fat of the orbit bulges into it and because the cheek skin is suspended by the ligament and the residual fat descends to the inferior ligament(s). Freezing, or otherwise destroying or removing the fat, only makes the problem of volume loss worse. Although you will look better immediately, you will eventually look much worse and need an increased amount of fill. Surgeons have also tried skin tightening techniques for years, but none other than surgical removal of the excess skin have been successful for longer than a few months. Instead of wasting your money on things that don't work, see a plastic surgeon who is an expert on fillers (including fat) and surgical blepharoplasty and discuss options that really work.