Will fat transfer under my eyes help with the wrinkles as well? (Photos)

You are right on. Fat will fill volume deficit but you will like need some botox to keep the deeper lines controlled. For the very fine ones, laser or another heat delivery device helps. I like the viva which is radiofrequency. 
Hope this helps.
Best of luckDrC 

Yes, fat transfer can help improve this area. Keep in mind, fat transfer technique is what's most important for successful fat transfer. I recommend water-assisted fat harvesting which gives pure fat to transfer and combine with platelet rich plasma (PRP) to get the most viable and sustainable fat cells. Fillers are an alternative and would allow you to try on the look before committing to a permanent option. I recommend getting a formal evaluation with an expert. Best, Dr. Emer

Fat transfer under the eyes can create irregular results after healing.  The fat may absorb in some areas and live in other areas making this treatment difficult to get consistent results.  Laser will resurface the skin and tighten it to some degree and can be used in combination with deeper fillers which are more reliable than fat. 

Yes fat grafting can be very helpful in this area.  It is best for filling in hollowed areas, and I wouldn't recommend it as a stand-alone solution for wrinkling in the area.  Resurfacing with a laser or a chemical peel can help with the fine wrinkles.  At your stage, I would recommend a trial of fillers to the area (which can give you and idea of what fat will accomplish) and consider some fractional resurfacing.  

 Fat transfer may help with hollowness, but the excess skin will not be improved by the fat transfer to any significant degree.

Fat transfer will reduce deep grooves but treatment of fine wrinkles will require a chemical peel or laser  at the time of your surgery and the use of quality skin care products after you have healed.