When choosing a home inspector, all those questions are appropriate to ask.
Beware of a home inspector recommended by the real estate peddler who is trying to get the house sold. The typical "real estate lady" (often a misnomer as regards the last term) wants to SELL THE HOUSE, and does not wish to have the sale compromised by negative findings from a home inspector. Thus, she (or a male agent if that is the case) will likely recommend the inspector who, in her view, is least likely to "make trouble for the sale."
I am very familiar with a home inspector in the Shoals area several years ago who was a very professional and objective inspector and who did a very thorough job. Numerous local real estate "professionals" (also often a misnomer, given the conduct of some of these folks) put out the word that this inspector could be a "deal killer." During the same period, one of their "favorite" inspectors was sued for sloppy inspection work and nailed with a big legal judgment.
So be careful. Perhaps a good way to proceed would be to ask the real estate peddler to name a couple of good inspectors and then to avoid them like the plague!