Home tests aren't 100 percent accurate, but they are actually very close to the accuracy of clinic tests if they are used after the missed period. I would recommend another test, done first thing in the morning (when your urine is more concentrated.) If in doubt, go to a clinic and get it done there. Remember, however, that it's pretty common to skip a period every once in a while.
Yes, you might be pregnant. Home pregnancy tests are not 100% effective. Anxiety is one of the many things that can cause you to be late so the best thing to do is take another test to find out for sure.