If a hair transplant is done with dense pack, does it typically take longer for the hair to start "sprouting"?

Before I get to your questions, I want to address density in hair transplantation. The density of transplanted hair depends on two things: the density of the surrounding healthy native hair and the hair transplant surgeon’s technique and ability. With hair transplants, we strive for natural looking, full hair coverage. Therefore, you’d want the transplanted area to fit in and flow nicely with the surrounding areas, matching the density. This will keep the hair looking natural (you don’t want to look like you had a procedure done). That leads us to our second factor in density: the surgeon’s abilities. If a surgeon’s abilities or experience are lacking, you’re going to get the best hair transplant s/he can give rather than the best hair transplant for you. This is why experience and techniques are so important when searching for a hair transplant surgeon. The punch size, placing by hand vs. robotic assistance, and involvement of the surgeon will directly affect the density of the transplanted hair. Over-sized punches, robotic tools, and absence of surgeons during the procedure are directly correlated to poor results such as a plug-like or thinly covered recipient zone. 
Now, to answer your questions regarding growth of transplanted hair. Transplanted hair will shed shortly after a transplant followed by new hair growth. What exactly this will look like and how long it will take differ's person to person. The hair must cycle through its specific growth cycle. If the density is done properly and blends in naturally with your current native hair, it should not matter if the recipient area is completely bald. 

It does not take any longer for the second transplant to grow. It can take 6 to 12 months after surgery to see results.  If you are concerned it is best to contact your doctor.

In general, no difference should be apparent in the growth rate even if the follicles are packed more closely together.  Good luck with your procedure!

That is a good question.  Generally, no, the growth should be the same regardless of packing.  Dense packing is often necessary to achieve natural looking results.  In other cases, it may be a wasteful use of follicles.  Do see a doctor with experience.

Regrowing hair start to sprout around 3-4 th months after hair transplantation and d it does not change whether the area had hair before or dense packing/loose implantation to the area .

Not really!  If the procedure is done right they should grow at the expected time after your hair restoration procedure. Here is a chart that walks you through what you should expect after a FUE hair restoration:

I am the doctor who first introduced the dense packing of the hair and the growth is identical to normal density hair transplantation