Forgive me, but if his identity papers are fake or forged and he used these papers to marry you - is your marriage legal?
Are the documents / ID in his real name? Did he use a forged or stolen ID to get into Canada?
Have you seen his passport - he would need one to get into Canada. What nationality is his passport and is it his? Are the documents / ID in his real name? Did he use a forged or stolen ID to get into Canada? Does he have a driving licence and is it in the same name? Have you seen his birth certificate?
If he needed a visa to enter Canada he should be able to show you it. If he hasn't then he may be here illegally under false documentation.
Without knowing his real legal name at birth it may be difficult to ascertain his true identity and his true reasons for being here. If you really want to check his identity then I suppose you could take his ID to the RCMP and ask them if it is real or reported stolen.
I have to say, as soon as I read your post alarm bells were ringing madly!
Have you known him for very long before he arrived? Did you invite him or did he just turn up - did you meet him or did he arrive at the house on his own? Has he talked about his family, friends, have you met any of them? Does he have relatives in the US or Canada? Has he received any letters or phone calls from friends or family? What name did they say?
What was he doing in the US and how long had he been there? Once the application for PR is in place he will need police checks done - will he pass them?
If "He said frankly, while residing in USA, that he was out of status there and
that it might have been impossible for him to come and marry me in Canada." how did he manage it?
I would hold off on applying for spousal PR until your suspicions and fears are allayed. Far better to have him annoyed at you for delaying than be obligated to have to support him for the next 3 years... and the suggestion that he return to his country of origin and then put in an application is a good one - if he is "genuine" he will understand that this has to be done properly in order for him to get PR status, and yes, it would be treated like any other application. If he is not genuine he will probably get really angry and possibly leave... you should be prepared and not give into emotional blackmail.
You will still need lots of proof that the relationship is genuine when you do apply - take this time to get it!
I apologise for all the questions, but these are questions you should be asking yourself if you suspect that he is in Canada illegally and under false pretenses.
I really hope that he is genuine and that all this worry is for nothing - but better to be safe. I am so sorry for your troubles.
NB If he is here illegally he will not be allowed PR status. See: http://www.cic.gc.ca/english/resources/manuals/op/op01-eng.pdf
5.23 Applications from illegal residents
R11 stipulates that applicants cannot submit applications for temporary or permanent resident visas outside their country of nationality—or, if stateless, outside their country of habitual residence—to a visa office in a country where they have not been lawfully admitted. Therefore, visa offices should return applications, plus any accompanying documents or fees, if they determine that an applicant was not legally admitted to the country where the submission was made. Applicants should be counselled to apply for a visa to a visa office where they have been lawfully admitted for at least one year.