Got a letter saying I'm pre-qualified for an FHA loan

[Art Maydan]

I'm in the process of getting pre-approved for an FHA loan to buy my first multi-unit and I got a letter in the mail from Marketplace MC saying they'll give an FHA loan with a "2.75%(3.14%) APR for a 15 year Fixed Rate!" or a "3.25% (3.46%) APR for a 30 year Fixed Rate!" It says my credit score can be as low as 580 and they'll pay my closing costs.

1. What's the catch with this? Are they advertising the best rate someone could get and the worst score you can have for FHA just to get inquiries and then will offer you loans with higher rates?

2. Why would a lender pay my closing costs? Might be worth paying a slightly higher rate if they'll pay my closing costs.

3. How did these people get my information and learn that I'm shopping for a loan?