Does a gestational carrier (surrogate) need to claim the compensation and living expenses reimbursement on their taxes?

Yes, you are considered an independent contractor. 

It is taxable income, but I am not sure that it is "self-employment" income, as opposed to "other" income.  The difference is that with self-employment income, you can deduct work-related expenses, but you pay income tax AND 15% self-employment tax on the "net profit" from the business.  If this is "other" income, you can't deduct expenses, but you only pay income tax and not SE tax.

I don't think surrogacy meets the tests to be considered self-employment.  I will check with another expert. @Hal_Al 

Interesting discussion here. 

https://www.taxpros.org/blog/surrogacy-fees-and-taxes/43453

I think it's pretty clear that it's taxable.

Another reference on the subject:

https://dimovtax.com/tax-implications-of-surrogacy/#:~:text=Payments%20Received%3A%20The%20compensation%20that,tax%20to%20the%20US%20government.

I'd hang my hat on the "one time gig" exception and claim it as other income and not as self employment.