Yarn breakage occurs at the sleeve joint section of knit underwear. Even when a humidifier is used, trouble persists. The sewing machine used is Company-make - (machine speed: 4,000 rpm). The needle used is a UY128GAS #9/65, the Yarn #50 spun.

Breakage of yarn occurs due mostly to the shape (thickness) of the needle used, with the tapered point of the needle and the blade having especially significant influences.
In respect to the yarn breakage referred to in your question, we infer that the breakage occurred due to the low level of strength of a single yarn in spite of the excellent expansion/contraction quality of the fabric used. As a result of tests we conducted focusing on the taper configuration of the point of the needle, we can report that a very remarkable result has been obtained with the J ball point. (Please refer to the attached table.)
However, since we could not fully resolve the problem by focusing solely on the point of the needle, we decided that changing the shape of the blade of the needle might be an effective countermeasure. As a result of tests we conducted using various blade shapes, we discovered that the "OSP" type is the perfect solution to the problem at hand.
We would therefore like to recommend the use of the UYx128GASOSP #9J needle to solve the above problem.
Please note that since all of the test results reported here were obtained under Organ Needle's testing conditions, it is conceivable that different results may be obtained when testing is conducted under your own particular conditions. In the event that the problem persists even when the UYx128GASOSP #9J is used, please try the UYx128GAS-KN #9 needle instead.

Skipped stitches are frequently observed to occur in a zigzag stitch of a ladies' foundation garment, with the effect of lowering productivity. The garment cloth is "Powernet" and the sewing machine used is Company's - (machine speed: 4,000 rpm). The needle used is a DPx5#11/75U and the yarn "Woolly" #60.

Considering that the material referred to is a composite yarn, we infer that said skipped stitches occurred because the level of resistance the cloth gave to the needle and the yarn varied depending on the place where the needle passed through the cloth producing, as a result, unstable loop formation. To solve the problem we think it necessary to bring the hook of the sewing machine as close to the needle as possible.
We would therefore like to recommend the use of a DPx134#11U of the DPx5 family, which is a long, deep "Cliff Scarf" type. Incidentally, we conducted testing on yarn breakage and conclude that the U-ball used by your company is the most suitable point shape.
Please refer to the attached test data.

The material referred to in your question is liable to cause seam puckering because it is made of extremely thin fibers, as is well known to you. As causes for the seam puckering referred to in your question, several factors can be considered. These include factors related to the feed mechanism of the machine, thread tension, and how the cloth is woven, among others.
It follows, therefore, that all conceivable factors should be taken into account when attempting to find a solution, including the sewing machine itself, the cloth, the threads and the needle used. As for the sewing machine, it is necessary to apply rather strong pressure to the presser foot, lower the height of the feed dog, make the hole in the needle plate smaller in diameter and thread tension weaker.
Use of cut-on-the-bias pieces of cloth to which a smoother has been applied will be found effective. (Use of a piece of padded paper is sometimes found effective.)
Thin threads less susceptible to contraction/expansion have also been found effective.
As for needles, the one with thinnest blade and smoothest surface must be chosen.
We recommend in this case, DBxINS #9ASU, which was found to be most effective.