There is no average lead time for being assigned because each MSC operates independently. If you apply for an assignment and it is not assigned by the time you need to know, simply cancel the application. Many systems allow you to do this yourself. If not, just send an email withdrawing.
I don't understand your question about how many shops you can apply for with a company. Some have restrictions on how many outstanding assignments you can have at one time, there is no limit to applications.
More companies than not will require your SSN or EIN. They need this to verify you are eligible to work in the U.S. and for tax purposes. Just beware of sending your SSN or any personal information in response to an unsolicited email. Verify the MSC is legitimate and use their website.
Equal rights for others does not mean fewer rights for you. It's not pie.
"I prefer someone who burns the flag and then wraps themselves up in the Constitution over someone who burns the Constitution and then wraps themselves up in the flag." -Molly Ivins
Never try to teach a pig to sing. It's a waste of your time and it really annoys the pig.