Yes, that Specialized Step through should work, but first you need to test ride it to make sure you're comfortable, and ride for as long as they'll let you, hopefully for at least 20 minutes to simulate your commute, that ride time should tell you if you're going to be comfortable riding it for that length of time. So make sure they get the bike set up to fit you before you ride off. Some people like those comfort bikes and others don't, so ride it first.
Before pulling your money out find another LBS that will probably be outside your neighborhood of course, and ride at least one more, that way you're not kicking yourself for not trying some others. It's like buying a car, you don't just buy the first car you see, some people might if they have a very specific car they want and find it, but most don't do that, they'll try out several cars from various dealers before they settle on a car, you kind of have to do that with a bike.
And the Step Through will handle the kind of riding you're wanting to do, look the kind of riding you do I sometimes ride those same conditions on a bike designed for fast road riding! What I mean is I use a bike with 23mm to 25mm tires on grass, hard packed dirt, hard packed gravel and have no issues, the tires on the Step Through are 45mm wide, that's at least 20mm wider of a tire than what I use for the same sort of riding, thus you'll have no problems riding on the surfaces you describe.
Commuting does present possible problems that are more frequent than a car, and those are flats. Make sure when you buy a bike that you learn how to repair flats if you don't already know how, if the bike shop will teach you great, if not what about a friend? if not that either then You Tube has at least a couple of dozen videos showing how it's done, watch as many as you can especially for the rear, and then practice on the rear where the mechanicals are so you get the hang of how to handle the mechanicals, how to pop on and off the tire, once you learn how to fix the rear the front is a piece of cake. You can prevent most flats by having the shop install a tire liner in both tires, those help reduce flats, but those are not flat proof but they will prevent most flats. Also when your tires wear out you can get better flat resistant tire along with the tire liners. In light of the possibility of flats you need to leave for work about 20 minutes earlier than you thought you would need to for riding in, in case of flat you have to have time to fix the flat so you won't be late getting to work, bosses won't like an employee being late to work too often.
Of course in order to fix you need to buy some tools, the bike shop should be able to select the stuff you need, like a pump that can handle high volume tires without stressing you out physically, tire irons, patch kit, and a spare tube, then a seat bag to put the stuff in.
If you're going to be commuting on surface streets you should consider getting at least a 70 lumen tail light, and you might want to consider the very nerdy dorky unfashionable but somewhat effective low cost (under $20) neon green, or neon orange (orange works better if you'll be riding where there is a lot of heavy foliage) safety vest with wide bands of reflective strips (these can be found at any home improvement store). You may also want to consider fenders in case you encounter rain so you don't get that brown rooster tail up your back and look strange at work.
You also need to maintain your chain at the very least, again You Tube videos to the rescue. But for all weather riding I like ProGold Xtreme chain lube; but before you apply the lube you have to clean the chain, and the easiest way to do that is to get a cheap sponge and some Dawn for Dishes, get the sponge damp put a little of the soap on, dampen the sponge some more and wring it out then get it damp again and start scrubbing the chain down by holding the sponge on the chain and spin the pedals backwards about two dozen times, rinse out the sponge real well, keeping it damp but free of soap wipe the chain down to get the soap off the chain, and then use a clean dry rag and continue to wipe the chain down to get any leftover water off the chain, let chain set for 12 hours and apply the lube per instructions.
Lastly, though not really, anyways you need to keep your bike maintained, so once a year take to where you bought it and have it maintained and checked out. I commute to work myself on a bike, it's not as bad as I made it sound, I haven't had a flat in over a year but you have to be prepared just in case.