+1(514) 937-9445 or Toll-free (Canada & US) +1 (888) 947-9445
You can contact them and request it, but it generally will not be approved without significant reasons - such as the birth of a child. Your COPR's expiry is tied to your medical exam's expiry. Delaying the COPR would mean re-doing medical exams.

You can leave Canada without a PR card, you can't re-enter. If you are staying wherever you are going in June for a good length of time, you can apply for a PR travel document, or have someone courier you the PR card if it is sent to you by then.
 
No, you don't need to "land" when you arrive in April. You just need to land before your COPR expires.