Sparks, try medicine or IT, that'll be fairly 'straightforward' for the purpose of US GC and ultimately citizenship. Why bother? Unless you get married to US citizen or win green card, it's pretty much waste of time.
I've read some of your posts, why would you want to stay in the US?
You're gonna be fine with M1 and J1 visas for flight training. Visa waiver or B2 for flying holidays. What more do you want??
Canada considers pilots, engineers, instructors and navigators as skilled migration qualifications/experience. That's much easier.
Just get trained, do your Canadian CPL while in the US (visit to other side of border to sit exams) or simply do conversion (VERY easy from US FAA, including type ratings - heli wise), sort the papers, get there on Working holiday visa (1 year open work permit as per Canadian visa/permit naming), do some networking, maybe bit instructing, if prepared well in advance and rotary wise with some good hours from J1 in the US, not so impossible to get heli job for a season - I know, you're FW minded. Just to let you know.
Then, you'd have company that'll give you job offer useful for residence permit/visa, stop by in Buffalo or elsewhere, go back to Canada (speeding up paperwork, otherwise apply few years in advance), pass 67 pass mark with skills, experience, job offer etc. That's it.
Much much easier than playing about trying to get GC through employment in the US.
As for NZ and OZ, they don't need another pilots, so unless you have some skills they want and do the legwork for residence, forget it via piloting..
Anyway, who'd want to work there for that money and with so much competition, right? :-D (yeah, for lifestyle, maybe)
Got it?