If you have a regular 28 day cycle, then you should have ovulated somewhere around June 18. It is still too early to test if that is the case. The earliest you could test would be around June 28-30. Tests used to be made that you could test the day of a missed period, however, the newer tests are much more sensitive and can be used as early as 6 days before a missed period. I think that maybe 3 days before a missed period is a safer bet with those tests. So, if it were me, I would try to wait until July 1.
No, it's better if you would know it earlier. For you to know what would be the right things to do.
Most likely. The most accurate time to test is the day your period is due, or when it's already late.