2. No, you need apply within 3wks from the date you stopped working (not earlier). Also the earliest time you can start to collect benefit is 8wks before dd. The first 2wks are waiting period, means you won't get any money at all, then your benefit will be dated back and kick in from the 3rd wks you stopped working.
3. Maternity leave is 15 wks for mom only. Parental leave is for the couple to take care of the baby, it's 35wks for a couple, can be started from the delivery date (no earlier). If you take it all, you will have 50wks leave altogether during which you can collect the benefit. You can not take both at the same time.
4. For contract or part-time job, you have to work at least 600hrs(not sure, maybe 700) within a year to be eligible for the UI benefit.