Once in US, you will need employemnt authorization to do any work on US soil, including work for your Cdn company.
But one need not wait for GC to have this authorization, one needs to apply for an EAD (i-765) as soon as your visa comes thru and you file I-485. This EAD allows you to do what ever you wish while waiting for GC.
so,
1) you can continue as you are now until you physically move to US, at which time you must stop working until you get EAD in your possesion.
2) Your business activity is your work. You have physically transferred yourself, so your "activity" will have by necessity moved with you. You will want to set up a business structure that conforms with US business and tax regs, since that is where you will be working. Since none of your work will be occuring in Canada, there is little advantage in keeping any Cdn aspect to it.
3) You must stop all work until you get EAD.
4) Anyone can set up a business in US, with or without GC; talk to a CPA to determine what business structure would be best for your case.
http://www.visajourney.com/content/k1flow