Firstly, you should understand that I do quite a lot of international traveling. In preparing my N-400 application, I had to list over 60 trips in the last five years.
Therefore, it is not easy to recall specifics for any one trip or those in a particular time period. Also, I only used an REP for a period of less than a year about three years ago.
With all that said, I will attempt to answer your questions to the
best of my recollection. These are based on my personal experiences with specific individual circumstances, and so may not be applicable to others.
I have never been sent to secondary inspection incoming to the USA, with or without my REP. I do not know if this is typical, but immigration is normally quite quick and I am asked standard questions. These usually include how long I was out of the country and my occupation.
Since I was a permanent resident, I stated my country of residence as the USA. However, I would let the POE officer know that I was currently living in Canada. This might result in follow up questions about where I was coming from and when I would be returning to the USA permanently. I would then answer with my temporary address and the fact that I was on a temporary assignment for up to two years. Again, I think the questioning along these lines was tougher before I received my REP.
My family, all of whom are US citizens, moved with me to Canada. Since I was renting before leaving for Canada, I had no home in the USA. All of my household goods were transported, including my vehicle. I was also required to give up my US driving license. Therefore, my only ties were my company, bank account and credit cards. My salary was being paid into my US bank account, and I filed all of my US taxes as a resident. However, none of this was ever asked about after I received my REP.
I was never asked for a letter from my employer after receiving my REP. In fact, even before receiving my REP, I never had to actually show the letter. This was despite it being "entered into the computer" that a letter should be shown the next time that I traveled.
I typically traveled across the border every 3 weeks with my REP, by both air and road. My absences from the USA after receiving my REP were between just 1 and 6
days. Prior to receiving my REP, my longest absences were between 30 and 40 days.
You may well ask why I even needed an REP with the above travel schedule. The answer is that I did not know how often I would need to return to the USA. In the end, it was very frequently, and I returned to the US permanently earlier than I originally expected.