Is needle mesotherapy FDA approved? (Photos)

Thank you for the picture.  I believe you should look at a couple of things for eyebags - either lasers for skin tightening or lower blephs by a board certified plastic surgeon.  In my practice,we use Titan eyes to help shrink and tighten the tissue around the eyes.  It works amazing and our patients are happy with the results.

You need to get a consultation with a board certified medical provider to give you options other than mesotherapy for your eyebags.

Hi there, 
It depends on which Mesotherapy products you opt for. Some are and some aren't FDA approved. The method of injection is not FDA controlled. 
That being said, I think you could consider other treatment options for your issues. 
For your pigmentation, I would suggest investing in a cosmeceutical skincare range. Dr Zein Obagi's ZO SkinHealth is a great option as it has amazing results with skin issues such as pigmentation, rosacea, acne, wrinkles, sun damage, melasma etc. 
Secondly, you seem to be lacking volume in your tear troughs. I would personally recommend an FDA approved filler in the area to restore this volume. This is a more high risk area, so please ensure that you are only having both FDA approved fillers and these are done by a medical professional (Doctor). 
You could then consider a laser treatment for any wrinkles or lines left. Something like clearlift is a really good option as it has no downtime. A more aggressive treatment would be iPixel Erbium or iPixel CO2. However, both of these do come with downtime. Although there is downtime, they provide much more extensive results than ClearLift. So there are plenty of options when it comes to laser too. 
I hope this helps and best of luck with your chosen treatment.