Salam doostan
@ms-nobody are you sure you need a supporting letter? I am not sure about pnp, but it is written in this link, you only need your nomination letter.
https://www.canada.ca/en/immigration-refugees-citizenship/services/work-canada/extend-permit/bridging-open-work-permit.html
I would say raise a CSE or call them and ask about it. Search on these forums as well and make sure you really need that letter.
As long as you apply for BOWP before July 1st, everything should be fine. Just email them and bug them otherwise they may forget about it completely.
As
@aasaf said if you apply for BOWP before the expiry of your current WP, you can work until a decision made on either of your applications, under something called "implied status".
Your SIN number will also be valid until the decision on one of your applications.
For medical plans, however, I am not sure as each province has different rules. I know that Ontario sometimes does not give health card to people who are under implied status. I have had friends who bought private insurance. But, it seems that they also don't know the rules properly. I have seen on these forums that someone mentioned you have to apply for it and after they reject your application, you can request for a review or something like that. Then, you have to go and explain it to one of their seniors and they they will give you for one year.
If I find the link to that post, I will share it here.