Hi syrust,
1. You could apply for In-Canada process. This takes abt 18 mnths to get ur PR. I'm not sure if your student visa allows you to work. If not you may have to wait for abt 8-10 mnths to get a work permit after you send ur PR application.
You could choose to send it to New Delhi(processing time 4-6 mnths). However looking at your circumstances(parents would not know/ no big wedding) I would think it's better to apply in-Canada as ND typically wants to see a big wedding with families, which is the norm.
2. Yes. When prooving your marriage is genuine all these will come in handy.
3. Well, if you are applying inland, this should not be a big probelm, provided you have enough evidance to show you both are a couple.
If you apply to New Delhi, this might be an issue.
4. No one would question abt why a Hindu and a Muslim are getting married in Canada

If you opt for a civil marriage, there won't be any problem.