Good luck, by the way. If your irregular cycles are driving you crazy, see if the doctor can prescribe some hormone therapy (progesterone most likely) to regulate them.
1) Probably. You might even be developing a luteal cyst, it happens sometimes after ovulation.
2) You will know by whether or not you continue to have normal periods.
3) No.
4) No.
Regarding question 3, first of all, this would not be week 1, but week 3, the way doctors count pregnancy. They begin the count on the first day of the woman's last period. But anyway, abdominal cramps at the time of ovulation are either going to be from mitelschmerz (a cramping from ovulating itself) or from something unrelated.
Regarding question 4, get to know the way it works. An egg pops out. A sperm finds the egg. The embryo is formed. It stays in its shell for 5 days while the cells split. Finally it hatches out of its shell by day 6. It floats around the uterus all this time, including after hatching. Perhaps it implants at day 6 or 7, perhaps it waits to implant for a few more days. NONE of this is anything you can feel, test for, or affect. Only once the embryo implants in the uterine side wall does it begin to signal the body and hCG gets into the bloodstream (and urine) and can be measured by a test. This is why they tell you on the test not to test too early. I mean, you can take a pregnancy test the day of ovulation or one day later if you want, but it will not show a positive, so it is wasting your bucks.