These are very good questions, and have stymied many an embroidery buff.
The answer to the first part of the question is possibly... I know it's vague, but it depends entirely on the type of hand embroidery thread you have chosen and what it will be used for.
It helps to understand a bit about threads and how they are manufactured. Threads for embroidery are comprised of twisted fibers called plies. Multiple plies that are then twisted together to make a thread. The thickness of the thread may depend on the number of plues used, or the thickness of the fibers. The individual threads in the plies may be smooth and lustrous due to mercerization or textured and wooly in their natural state, depending on their fiber content and the thread's intended use.
These plies and threads are either "s" twisted or "z" twisted, referring to the direction of the twist (to the left for s and to the right for z). Note that manufacturers do not label their threads with the twist direction, because the direction of the twist has very little (if any) bearing on stitching.
Threads and fibers for embroidery may be single-strand threads like floche, or multiple strand threads, having groups of threads - usually 6 - combined to create a multi-strand embroidery thread such as six-strand embroidery floss.
Nap, on the other hand, has nothing at all to do with the twist, but instead refers to the direction the fibers are running in each of the individual twisted plies. Every thread has a certain degree of nap. For example, floss and pearl cotton have a low nap that is not visible unless magnified, while wooly threads such as crewel wool or fleecy threads have a high nap that is obvious. Low naps are not noticable or worrisome on most threads for embroidery. However, with textured embroidery threads that do have a noticable nap, the stitcher needs to determine whether or not the nap of the fibers will affect the stitching.
Here's an easy way to look at it; Think of the nap like you would think of your own hair. If you comb in the direction of growth, your hair is smooth and tangle-free. If you comb the opposite direction, towards your scalp, your hair becomes matted and ratty. Fibers with a fuzzy, stringy or otherwise textured look to them will indeed have a nap and this should be taken into consideration when stitching.
Stitching with the nap - pulling the thread through the fabric in the direction of the nap - avoids thread damage by keeping the fibers combed in their natural direction. Stitching against the nap results in breakage, loose fibers or fuzzies, tangles, matted fibers, and damage to the thread as it is combed by pulling the fibers through the embroidery fabric.
If you are using a pull-skein of floss, a hank or ball of pearl cotton or a spool of silk, chances are your thread has no nap. This thread can be used in either the direction it comes off the skein, or the opposite direction without any issues. While any thread has a certain degree of nap, the majority of cotton and silk embroidery threads do not have a noticable nap that would affect stitching.
However, fuzzy wool threads, velvet thread and many other specialty threads with decorative or 3-dimensional fibers do have a nap. Examples of these threads with and without naps are featured in the photo, and include the velvet, loopy, or fuzzy threads that have a high pile and do indeed have a nap. Threads with naps are often used to represent hair, santa beards, animal fur, pebbles, metal, rippling water, tree bark or other dramatic or textured element in embroidery patterns or designs. When working with threads with a nap or directional element, the thread should be stitched by pulling it through the fabric in the natural direction of the thread's nap to avoid damage.
For the second half of the question - How do you find the nap? In most cases, the nap can be seen with the eye. You can also detect the nap by running your fingers along the length of the thread. If your fingers run into resistance along the thread, you are going against the nap. If your fingers run along the length of the thread smoothly, you are running your fingers with the nap. You can test this method on a scrap of velvet, thick corduroy, terry cloth or other high-pile fabric. Run your hand in varying directions on the fabric to feel the nap. When running your hand with the nap, the velvet or corduroy fabric feels smooth to the touch and shiny, while running against it creates resistance against your hand and give the fabric a disheveled appearance. The same principles apply to threads with naps.