My wife (an elementary school teacher) originally worked for employers (school districts in Texas0 who were in the Social Security system. Then, she switched employers and works at a school that opted out of the Social Security system years ago. They do not take social security taxes out of her check and they do not contribute social security taxes under her social security number. In her early work inside the social security system she earned more than 40 quarters. Under her new employer she contributes to the Texas Retirement System in a manner that is remarkably similar to Social Security taxation. My 2 questions are, (1) "under what circumstances can she draw social security retirement benefits?" Also, I have heard that she could draw survivor's benefits from Social Security based on my social security eligibility, but only if she wasn't retired and assuming of course that I had died. (2) Is this correct?

Here is the best source: Social Security Online - Retirement Benefits.

It appears to me that she has the necessary number of quarters and collect upon reaching either early or full retirement age; also, yes, she can collect from your benefits if you die first. Her ability to collect as a survivor has nothing to do with her employment or contributions.