1. Do you mean to say that at the port of entry, if the officer would like to report a person for not meeting residency requirement, he will give you the paper work right there and then? What does it mean if the officer just let you in (without any paper work) after secondary inspection -- does it mean you are still a PR and your status is fine (even after the expiry of your PR card)?
Yes, they should give you paperwork right away. If they let you in without paperwork, it means you are still a PR, not under investigation and your status is fine.
2. If a person who cannot fulfill the residency obligations for the first 5 years as a PR but manages to get into Canada (after being sent to secondary inspection at immigration at port of entry) before the expiry of the PR card and then stays in Canada for a full 3 years or more, can he/she apply for PR card renewal and citizenship with no problems?
Yes, I quoted the manual above. If a PR arrives in Canada after many years away, gets in without getting reported, stays for 2 years, then applies to renew PR card, they can only look at the last 5 years, they must admit he meets the residency requirements for that period and renew his PR. After one more year, he can apply for citizenship.
3. Is there somebody in the forum who was successful at applying for PR card renewal or Canadian citizenship after staying in Canada for 3 years or more but initially cannot fulfill the residency requirements in the first 5 years as a PR?
I have seen people post about that.
4. Is there a way to know if one is subject to immigration investigation even after the immigration officer at the port of entry let the person in to Canada before the expiry of his/her PR card?
I don't know if there is a way to see that. If they don't send you any letters, you are probably not under investigation but if you start enquiring into it, they might take it as a sign that you should be under investigation. If you stay 2 years and nobody has contacted you about being under investigation, then it is not very likely that you are so you would be safe to renew your PR card at that point.