Does the lenders run the credit check to get mortgage rates ?

I am in the process of buying a home and did like one of them who did take my offer. Now coming to the actual loan process.

I see several website who give me a online quote as how much my interest rate would be and other things..

1) Will they all stick to that rate what they have quoted online ?

2) Or do they have to actually run my credit history and then can give
me final rate ?

3) If I have to shop around, does everyone runs my credit history ?
Does it effect my FICO score ?
Will it drop my score if currently i am around 790

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