Read your application guide and document checklist and you will see what forms you need.
When applying as a skilled worker, it is not necessary to take the IELTS and if you do, it's not necessary to take it before you apply to Sydney. If you pass Sydney, then you will be asked for documents and you will have 4 months to take the IELTS then. However, it is recommended unless English is your first language that you take the IELTS and it's also recommended that you take it before you apply in order to properly calculate your points. It depends on how much you need those points. Sometimes it happens that people are banking on getting full points for English and by the time they take IELTS, they get a lower score than they expected, end up with less than 67 and lose their application.
In your application guide, it explains how you should send your application to Sydney, NS. This is the office that decides if you are qualified to apply. If you pass them, your file is sent on to your visa office. You can select your visa office on your application as either your country of nationality or the USA since you are legally admitted with at least a 1 year visa. Whichever you decide on, you have to read the instructions and follow the guidelines for the visa office you select.
They may contact your employer to verify your data, yes.
