There's absolutely no problem staying with your partner in Canada while an outland app is being processed. We mistakenly thought that, which is why we filed inland, a decision I long regretted.
I'm not much help with the country of residence bit. Like I said, I listed Canada, as that was where I was physically staying at the time. Caused me no issues, though I'm not sure it was correct. Do you have a "permanent" U.S. address you can use? For instance, I use my sister's address as my U.S. address for anything that requires one. I'm pretty sure, though, that you can list the U.S. as your country of residence but list a Canadian address. You are a citizen and resident of the U.S., even if you want your mail sent to where you're staying in Canada for now.
I'm not much help either with the border part. I crossed into Canada by car with my husband (we were kind of lucky - he's military, so all of our household belongings were being moved professionally and we only had suitcases with us). I told the Immigration officer (who was really nice to me, btw) that I was traveling with my husband to his new posting (true), that he had to leave that new home in Québec to spend around 5 months in Ontario for a course in just a few weeks (true), and that I was planning to, when we got the exact dates of his course, fly back to Oklahoma to stay with family while submitting an outland app for PR (false). He seemed to know exactly what I was doing (after I said I planned to fly back, he said, "And THAT is what I was waiting to hear") and even told me I didn't need a lawyer for the process ("I'm not allowed to advise you on this issue, as it could threaten my job, so I'm not telling you right now that you can absolutely do the app yourself. You hear me? I'm not saying you really don't need a lawyer." I told him I was not hearing what he was not saying loudly and clearly.) We had absolutely no trouble, despite the fact that I'd quit my job and had no lease or anything to show at the border. However, it depends on the day and the agent at the border.
If you take a letter as extra precaution, I'd think "To Whom It May Concern" would suffice.
I just wanted to be certain because on the checklist it said to make sure original copies of the forms (including the IMM0008DEP) are included on the package. Haven't read anything yet that says not to include forms if not applicable to the applicant.