Does someone review your credit report when applying for a mortgage?

Of course there is.  When applying for a mortgage or loan, this is a standard procedure and you are not allowed to be there to explain anything.  They determine your mortgage payments based on your credit rating and your interest amount.

All inquiries within a 15 day period are considered 1 inquiry

Your late payments from 5 years ago, have almost no impact on your current credit score.

You don't need to explain anything.

Yes they do, and they will contact you if they find discrepancies.

Yes.

The final stage is called "underwriting" and an actual human being looks through all of the documentation such as your bank statements, pay stubs, and your full credit report (not just your score) and makes the final decision on whether or not to authorize the loan.

Its common for last minute questions to pop up during this phase, and as long as you have valid answers and proper documentation it should not stop you from getting a mortgage.

When you first meet with a mortgage agent be open and honest with them about your situation.

The inquires for the car loan shouldn't be an issue. This is common and the scoring algorithms actually only count this once. Its common to shop around or run credit with multiple banks when looking for a loan, so multiple inquires for the same type of loan within a short period of time will count the same as if you only had one.

The late card payments are probably old enough that they aren't going to be a deal breaker but again make sure you tell the mortgage agent about it upfront so you don't end up with unpleasant questions from the underwriter 2 days before you're supposed to close on a purchase.

Your score takes nearly everything into account. But eventually a human will look before closing if it gets there.

Multiple inquiries for like a car loan are only counted as one if they are from a short period of time.

5 year old late payment is unlikely to have a huge effect. A new car loan might.

August of 2020 is not 5 years ago.

Yes, an actual person looks at your credit report.   More than one, actually.

Lenders don't care about explanations for late payments.   They care about whether you pay on time or not.