One should never ever stretch before doing weight training, as it will reduce the power of your lift and will cause loss in stability and this could lead to injury.

People who say dynamic stretching is good probably refer to activities-like baseball, etc., where you usually just warm-up by pitching a ball back & forth to the teammates or in sports-like, martial arts where stretching can prime the muscles in order to get them ready for use.

if this is the case, stretching is good but denitely not before weighttraining.

One more question which might come to our minds would be that can we stretch between the sets or can we just sit and rest between stretchs?

I would say no, as the idea of stretching between the sets is that you are cleaning your muscles out of the lactic acid when you do a stretch.

Imagine stretching of our biceps or chest to be similar like "ringing out a towel" that is soaked with some water. Therefore, if you do this, it presents same problem as written above by me, i.e., reduced power and losses of stability. It depends on your program and this is not such a big deal, but there is always a better side.

If you talk about between the sets, do some mild aerobic activity. Define your path through your gym between your machines and then walk up and down at a fast pace. When you do this, it cleans out much more lactic acid and will bring in much more oxygenated blood to the specific muscle you are training.

So it is for sure that do not ever just lie or sit and do nothing. This can be considered as the worst form of relaxation technique.

One more thing to add that some difference between all your exercising activities must be taken into account.

It will be much more stressful for youself to move from a fully relaxing position to a suddenly running position, rather than moving from a very slow jog to running position. Akin to this, it is much more stressful on your body to move from running too fast to fully relaxing. Least but not last, I would say it is much better to move from a fast paced run to a very slow jog and now to a fully relaxing position. Again, This is another reason as it is just not good to sit between sets or running or jogging and to do nothing between the sets.

Never, unless and until by running you mean, "all out fast as you possibly can sprint, then no.

You need not to just warm up. Jogging is considered as a warm-up activity. How one can warm up before a warm up? It is totally senseless.

Do you think that jerking while doing streches will cause an injury to your body?

Yes, it is true.   Even bouncing while doing stretches or bouning at the end of the stretch will cause an injury. If you are gently moving back and forth, it is okay but never go further than this.

You might be wondering if warm-up and cool-down can be done activity-specific rather then doing the same type of stretches again and again?

Warm-up and cool-down should specifically be done depending on your activity and never with the stretches.  Stretch is not a good way to warm-up.  If we are doing heavy set of bench-presses or any gym machine exercise, we can warm up by starting with very light weights and press only 15 times. Like this, and then we can add some more weight and do around ten or 12 repetitions.  After that,  we will add more weight and do only five or six repetitions. At last, we can add more weight and do our heavy set of three or four repetitions.