If you tell the consultancy that you have GC, they will most likely make you pay before the training starts. That is because consultancy do not have to worry about your papers and any consultancy will take you in .So either make the payment first or you got to start lying from begining like you are on other visa , probably opt . You can call almost any consultancy and most of them do provide training and accommodation ,but keep in mind that training itself can not make you very competitive for 5-6 years of profile. Also the accommodation they provide will probably be shitty.
QA these days are pretty intense as manual testing wont give you that much of opportunity. Automated testing requires coding skills. So , if you do not have any IT background, I suggest you to look for BA . If you have decent communication skills, you should be fine.