It IS the roast that is the difference. The only real difference in the beans is that some beans taste better at a higher roast than others, so they are more appropriate for espresso. Your Italian grocery coffee company may be using the espresso label for marketing purposes, but in general, espresso coffee beans can be the same beans that are used for "regular" coffee, but roasted to a French or Italian roast level, which is darker than City or Full City.
Since the advent of Starbucks, many roasts are much darker than they used to be. Dunkin' Donuts coffee, which is a Full City roast, used to be the norm, but now a French seems to be what you can buy.
I roast my own coffee and take it to just into the second crack which is, generally, a Full City roast...a point where the character of the coffee predominates rather than the flavor of the roast. There is more information about roasts at Sweet Marias where I buy my green beans, and reading through the site will give you way more of a coffee education than you probably ever wanted.
So, yes, you can use the coffee you have to make brewed coffee. It will probably be roastier than you would normally have, unless it is just a marketing ploy, in which case it will taste normal. Consider how long you have had this coffee; if it has been shelved for a while "normal" probably won't be all that great, since freshly roasted coffee is, generally, way better than old coffee. But as long as the oils aren't rancid, it is more likely just going to be bland.
You list the main differences already (and yes they can be both made from the same bean or blend).
The name turkish coffee refers to the preparation method ... and the grind/granulation/coarseness is adjusted to the method.
Coarse ground coffee - Turkish preparation:
I come from a culture where Turkish coffee is brewed in almost every
home, so I always preferably go for it. And sometimes not having the
fine turkish-style coffee powder at hand (living abroad) I have made (read: tried to make)
"Turkish coffee" with a coarser ground coffee powder many times -
the taste is often quite alright, as you can compensate for the
surface (exchange area) with adding a bit more powder or letting it
sit a bit longer. However, as for this preparation you leave the
powder in the coffee (also while drinking) the coarser coffee does
not sink easily ... so you get it all between your teeth, if you
don't filter it out somehow (which is then not the traditional
Turkish coffee anymore).
Turkish-ground coffee - filter preparation / French press:
On the other hand using a French press, the fine Turkish-ground
coffee escapes through the mesh. This is not too bad, as the super fine coffee will sink and you will end up with coffee brewed in the carafe (like the "mud coffee" or kafe botz that they drink in Israel), with the unnecessary addition of making the French press dirty. Used to prepare filter (drip) coffee, the
finely ground coffee clogs the filter paper. (being the adventurous
type I have experienced all this already, but wouldn't dare
recommending it)
In both cases you get coffee as a result after some hassle ... and even the desired taste might be reached ... but using the right coffee grind with the right method makes everything way easier.
The best is if you can grind your beans yourself (in Europe you can find public coffee grinders in many supermarkets and stores) ... then you can always achieve the desired coarseness for the right method of preparation (drip brew, french press, espresso, turkish,...).
Some additional notes:
You mention arabica - it is the most widely cultivated and
commercially used species of coffee (Coffea arabica) - so you will likely find it used in most preparation methods; the
other species that you sometimes get is called robusta (Coffea
canephora) and has a much stronger bitter flavor.
Not only the specieas, but also the roasting is very important for the flavor. So with the same bean you can get very different coffee (from a very strong dark roast to a very light roast). In some places I have seen turkish coffee prepared by quickly roasting the dry coffee powder in a cezve before boiling water was added, which gave it an extra toasty aroma.
In some places, turkish coffee will be made with the addition of spices (e.g. cardamom), which you can find also already in some ground Turkish/Middle Eastern/Iranian coffee blends. This will quite likely make a big difference in flavor when compared to "French" coffee (or to some drip-coffee blends that are sometimes made with the addition of hazelnut or vanilla flavors).
Best Answer
Yes, It's a Moka Pot. Despite the name, you can't prepare espresso coffee in that pot.
The coffee that you can prepare with the Moka pot is not even close to an Espresso. The pressure is key to extract the coffe aroma and to produce the cream. The Espresso requires 7 grams of coffee powder and high pressure 90°C water to produce 25ml coffe (cream included). There's no way to do such a thing with a Moka pot.
With a Moka pot, you will get a coffee, which is not Expresso but is also a coffe which is widely drink in Italy (at home, usually). It's different from Espresso but, in my opinion, good. It has it's own taste. The closest thing to an Espresso you can get, if you woUld like to go that way, is a small amount of coffee, but you won't get the same flavour, nor the density.