For example say someone has sensitive skin and they can feel the needles, Is there a point that can numb the pain?
There are points for pain, but not quite in the way that you are describing. In my opinion, needling pain is never due to the patient and always due to the practitioner. With appropriate needling techniques you should never have a patient that cannot tolerate needling.
While personally I think it makes little difference as technique far outweighs any benefit - there are different qualities and, of course, diameters of needles. Some patients may respond better to the smaller Japanese style needles (i.e. Seirin) that have a silicon coating on the needle for smoother insertion.
If a practitioner cannot insert the needle by hand without causing discomfort to your patient, just use the insertion tube and the patients should not have any issues.
Are there specific points for pain receptors? That seems like one is treating pain with an analgesic (Tylenol) rather than treating the underlying reason one has pain.... but are there the morphines/ pain killers points in acupuncture?
Yes and no. There certainly are acupuncture analgesia points (both auricular and body), but clinically they would rarely be used exclusively and some are better in conjunction with other points for certain conditions than otherwise. One theory is that all acupuncture points elicit some kind of endorphin/opiate response, hense the relaxation effect of acupuncture and it's overall usefulness with pain. Some points appear to do so more than others and consequently are used more in pain cases. One auricular point used commonly for pain reduction is Shen Men. Not an analgesia point, per se, and used for other conditions, but the effect on pain is considerable.
You might find the following research studies interesting (there are many more, but I think these two are fairly comprehensive):
Acupuncture Anesthesia and Analgesia for Clinical Acute Pain in Japan (PDF Version)
Auricular Acupuncture for Pain Relief after Ambulatory Knee Arthroscopy—A Pilot Study (Pubmed Online)