it is legal if an offer letter states that company will not pay if employee/contractor is terminated from project within first 10 days?

If you are an actual employee--not an independent contractor--then you must be paid for all work done, even if you are terminated for performance issues within some defined time frame. The law (e.g. Fair Labor Standards Act; state employment laws) are clear that employees must be paid for all work done. So if terminated the morning of day 3, you'd still need to be paid for the work you did days 1 and 2.
However, that's ony if you are an actual employee: the labor laws and their protections do not apply to contractors. Payment of independent contractors is, as the term "CONTRACTor" implies, governed by the contract or agreement, and whatever the parties agree to is legal: so a contractor who agrees (which agreement can be shown by taking the job knowing the terms and conditions on it) to some limitation on payment or waiving/giving up payment under certain circumstances is held to what he or she agreed to.
However, sometimes people are called contractors when they are really employees: the law does not care what the parties *want* the position to be, but looks instead at what it *is*. If a "contractor" is really an employee, he or she must be treated and paid as an employee (including also the employer paying its share of social security and Medicare taxes, making unemployment contributions, providing whatever benefits its other employees get, etc.). If you are called a contractor but function as an employee, go to the U.S. Dept. of Labor website, look up "independent contractor," and compare the requirements to be a contractor vs. your position. (The IRS also has information about this on its website.) If you believe you were "misclassified" as an independent contractor when you are really an employee, contact the state or federal department(s) of labor about filing a complaint.