Hi,
I know this is an older thread, but am in a similar situation. I studied for an year and received my work permit for an year which is supposed to end in June 2019.
I started my job on the same day of receiving my work permit.
First Question: Is there any way that I can apply for a PR under CEC and stay in Canada until I receive my PR?
Yes, and that way of course is to apply for a new work permit. You can either ask your employer to apply for an LMIA or look into other programs (e.g. IEC programs).
Second Question: There were a couple of people in similar situation? Did anyone face any problem if they applied under CEC after 11 months of work experience?
You are not eligible to apply with 11 months; assuming you get an ITA and apply anyway, you will be refused.