Don't think that border hopping is a real solution -- U.S. customs officials do not care for it at all. Here are things that are going to trip you up.
1) When you cross the border the first time after you have been in the United States for 6 months: "What is the purpose of your visit?" If you answer, 'Visiting,", they are going to ask you where you normally live. If you're living in the United States, your answer to this will either be a lie or the truth. If it's a lie, that will come out under further questioning when they ask where you work, when you're going home, etc. If you tell the truth, they are going to ask what visa gives you the right to live in the United States. Unlike Canada, U.S. border officials do not consider a 6-month visa to be an automatic right to stay for 6 months -- it is six months which you have to fulfill the original stated purpose of your trip. What was that? Did you tell them that you were going to live for six months, and then try for another six months? The best that can happen is that you get in; it's more likely you'll get in with a large red flag on your passport file; the worst is that you get refused and your visa revoked on the spot; and actually, there is probably something a lot worse than that, but I don't know exactly what it is.
2) There is no such thing as border hopping, because the U.S. regulations are 6 months in, 6 months out. If you go 6 months in, one week out, go back in, you might get six months, but it is at the whim of the officer and nothing else. Your plan is actually to ask for mercy at the border; which is another way of saying that your plan is to hope for a non-jerk border official. This is a dangerous bet on the U.S. border.
By the way, I speak from experience -- the FIRST time my wife and I went back in, she was aggressively questioned, and we got through by telling the truth from the get-go: that we knew it was a mistake for her to live with me for 6 months, that we'd been up in Canada arranging to move back, now we only needed two weeks to pack our stuff and go; and if they couldn't give it to us, well, we understood that and I would put her in an apartment and go back and move myself. We have a young dual-nationality child that probably gave our family a bit of legitimacy in the eyes of the officer (not implying that that's fair). They gave us three weeks, and warned us that the next time she crosses the border we would need to carry substantial evidence of being settled in Canada. You don't sound as if you have the financial resources to prepare yourself for a sudden inability to go home, so think twice before risking it. The crossing I went through had two lines, and if I'd chosen the other, who knows what might have happened? No fun at all. My wife has a 10-year, multiple entry visa, by the way.
3) They aren't going to take a PR application as proof that you're planning to go back to Canada -- "What if your application is refused?" will be the first question. ""Do you have a home or job in Canada?" will be another question. "Why aren't you living there?" will be a third question. "If you want to live in Canada, why aren't you living there while you apply?" is a possible fourth question. All of these answers will reveal weak ties to Canada -- and weak ties to Canada, plus a relationship to an American, plus living in America, all suggest GREEN CARD APPLICATION. Border officials despise foreigners who sneak across the border and then apply for a green card; which gives them the right to stay and to work while it's being decided. You are going to look like a high risk for that.
4) How are you going to prove that you are applying for a PR card? It's weak beer even if you are, and they aren't going to care about some receipt from CIC. For all they know, that receipt is from before your application was rejected. They might also know that if you are not resident in Canada, it is not so easy to be a sponsor.
5) They won't care that your American partner is planning to move to Canada, but they will care very much that you look like you're moving to the United States. Remember, the PR application is hypothetical, but you, at the border, after a 6 month stay and a two-day exit, are real -- and you're trying to get in. And you've got all sorts of reasons why you won't stay, but you don't have a job and you don't have a home and you don't have a plan for if you can't.
6) And none of this even gets into the REALLY jerkish questions that could come up: how long have you been together? Suppose you break up, how do we know you'll go back to Canada? How are you supporting yourself in America? Is it legal to be getting your financial aid while you live in a foreign country? Suppose Canada cuts you off? And so on.
Sorry to be cheerless! I've traveled a lot and lived in many places, and planning technical overstays in the United States is dangerous and short-sighted.
