A lot of these sounds like typical headhunter scams, specifically by getting to know your previous employers, some headhunters make direct contact with them in hopes of scoring more clients; and by knowing your previous paychecks, they have massive leverage over you in payment negotiations (employee salaries are not comparable to contracting daily rates).

Reality check: do you have an offer on the table? If no, let them know these details are available once you do. If yes, consider including redtape into your contract days.

Also, and this might sound obvious, but Know Your Customers: specifically perform at least a handful of searches against both the contracting company, the staffing company, all the relevant decision makers, and everyone you know the name of so far. Do they look legit? Are there other contractors on their payroll? Can you contact them, and ask their experiences?