what will help reduce these droopy areas at the sides of the mouth? What are they called? (photo)

The muscle that pulls down the corners of the mouth is called the DAO (depressor anguli oris). By relaxing this muscle with Botox, the creases often soften. Additionally, we also use hyaluronic acid filler such as Juverderm or Restylane to fill in any residual depth of the creases. Unfortunately, there aren't any facial exercises that can reverse this problem.

The droopy angles of the mouth are called "marionette" lines or "sad" lines. While they often become more pronounced in a person's forties, I have had many cases in my career of these lines appearing even in the early twenties. Marionette lines typically result from a combination of excessive downward pull of the depressor muscles to the sides of the angles of the mouth, known as the depressor anguli oris (DAO) muscles, and shifts in facial fat fat compartments in the cheek region above the sides of the mouth. Fortunately, the marionette lines can be treated quite easily by addressing both problems. The dynamic downward pull of the muscles can be lessened by the injection of a few microdroplets of a neuromodulator, such as Botox, Dysport or Xeomin, and the static component resulting from the changes in the fat compartment above can be restored by injecting a volumizing filler, such as Perlane L, to lift and buttress the area immediately below the droopy corners. The earlier one starts treatment the better to improve the appearance and even prevent progression. Given the fact that we have emphasized that part of appropriate treatment is to soften muscular activity of the depressor muscles, not only would facial exercises not be of value, they theoretically could prove counterproductive. 

Your weight loss has made the change apparent.   Additional weight loss will worsen their appearance.  You could consider filler, such as Restylane but facial exercises will do absolutely nothing of benefit.

The down movement of the angle of the mouth is called marionette lines.  They are caused by the pulling of the mouth by the depressor of the angle of the mouth (DOA).  The constant pulling will create a volume deficit in the triangle located under the sides of the mouth.  For the best results Botox the DOA and fill the marionette lines with fillers.

These are the beginning of Marionette lines, a line or crease that forms by the side of he mouth and angles downwards.
A hyaluronic filler like perlane, resylane, juvederm, or boletero, can help reduce the crease. There are no facial exercises that will
make it better and yes, weight loss can cause relaxation of teh facial skin creating the line/crease.

injecting Botox or other neuromodulators such as Xeomin or Dysport, came improve the position of the corner of the mouth.  He don't want to inject actually at the corner of the mouth but more at the base of the muscle draws it down.this muscle is called the depressor anguli oris.  Sometimes people refer to it as a DAO injection.in any case UR a bit on the young side so proceed very cautiously.  weight loss may have something to do with it and that if you lose volume in your cheeks or midface it may allow your mouth and lower half of the face to appear more drawn.He should get a consultation before proceeding with an injection and if you do make it very conservative.

Based on your photos, you have developed 3 lines on your face, the tear trough, laugh lines, and marionette lines, and between the lines appeared islands of raised skin. These lines are primarily caused by sagging on your mid face and lower face. The best solution for your case I believe is S-Lift or MACS lift or Spring Thread Lift.Also alternative and acceptable methods to correct the lines are fat transfer or filler injection.