What is the cost basis of a gift of equity purchase?

[Loaner]

I plan to take over my parents house by paying the remaining mortgage. They bought this house for $100,000. This was their primary house and will be our primary house.

FMV $550,000

remaining mortgage $350,000

gift of equity $200,000

Questions:

1. There will be no capital gain for my parents as this is their primary house, correct?

2. What is the cost basis for me when I sell this house, $550,000 or $350,000?

3. If my parents still live in the house with us after the transfer without paying rent, is that going to be an IRS issue/red flag?

4. What is difference between the above setup and purchasing this house for $350,000 without gift of equity?

Thanks for your help.