Setting up company in USA
Thank you in advance,
You can absolutely own a company in the USA and manage it from Russia. To open a bank account however you would need to travel to the States (you cannot open bank accounts remotely any more).
As far as choice of a state to register the company in I would recommend you to consider either Delaware or Wyoming, those two are very popular incorporation destinations for import-export companies owned by internationals. Keep in mind that you will be able to open a bank account in almost any state, not necessarily in the state where you plan to register the company (for example you can open a bank account in New York for a Delaware company).
As far as the choice of entity - I would suggest you to go for LLC and not corporation. In your case with LLC you can pretty much do anything you could do with a corporation, and having LLC would make it easier to maintain the company in good standing (less formalities).
Now to your other questions: unfortunately we cannot give any immigration related advise, but from what I know ownership of a company does not really help in obtaining work visa. You should consider consulting an immigration attorney or specialist, since you might have certain advantages in obtaining business types of visas (but again, those questions are better addressed to an immigration law specialist).
And finally, your bookkeeping and taxes should be handled by professionals (preferably here in the US). When placing an order with us we offer an opportunity to receive a free tax consultation from a licensed tax specialists that we partnered with for that very purpose.
I hope that helps.