Probate sale indicates that the property is being sold as part of the execution of a will. Depending on how the will was written, the sale may be contingent on the approval of multiple beneficiaries. For example, if a person dies and leaves his home to his 5 children in equal shares, then all 5 children would have to approve a sale before it goes through. I would expect in your case, however, that the beneficiaries of the property in question would have agreed upon the sale price before putting it on the market.
The probate as a whole process can be extremely protracted, but I don't think the sale the property itself would be necessarily slowed down, unless a dispute among the beneficiaries arises.
Like Bran9192 said, you should hire an inspector at your own expense, just so you know what you're getting into. I don't think there is any requirement for anyone to inspect the home prior to a sale, but it would be unwise not to do so. But you shouldn't expect the sellers to correct any deficits, as clearly indicated by the "as is" clause.
With regard to the appraisal, since the sellers want cash (i.e. they don't want someone to bring an offer with a financing contingency), then an appraisal isn't needed. The appraisal is important when you're trying to get qualified for a mortgage and the lender wants to make sure that they're not going to lose a ton of money by approving a loan for a grossly overpriced house.
