1a. No, a very sensitive test is not necessarily very good at ruling in disease, unless it is also very specific.  For example, if a test was 95% sensitive but only 60% specific, and 10% of patients had the disease, you get the following 2 x 2 table:

Thus, only 19 of 91 patients with a positive test have the disease.  That's not very good at ruling-in disease!