I am not the person to give you a definitive answer on whether or not 4 months time will bring partially cured firewood up to "fully" cured. I suspect much of that has to do with the relative humidity where you live. Wood would likely cure faster in Arizona than it would in Louisiana. Make sense?

Firewood WILL cure in the open air (we have a couple of stacks that aren't under cover right now, just a tarp); but there are a few things to consider. Make sure you stack your wood on an old pallet, you don't want the wood to be in contact with the damp, moist ground. You want air to be able to circulate over and around the individual pieces. You should cover the stack, but LOOSELY and mostly over the top. The object being, you are trying to keep the rain/snow from saturating the long, exposed surfaces of the top pieces repeatedly.

You can leave the bark on. Actually, it will help give you an indication that the wood is drying. As the wood dries it contracts and the bark will become looser. Also look for telltale splits and cracks on the butt ends of pieces.

Is 4 months long enough to "season" wood? No. But it will burn just fine in your stove.
Seasoned wood becomes gray. That's the "good" stuff. But not many people around here wait a year to burn the wood. If you get your wood now it'll be ready to burn this winter. But get your wood from somebody that sells firewood. Because they usually fell the trees in the winter and stack them up. Now is when they're cutting those trees to log lengths. The wood will look like it's not good enough to burn, but it's just fine.