It's staying, too muh work to close it up. Just wish it would swing open, instead of into the house. Can I simply just switch the hinges to the other side of the door to do this?
You can certainly switch the hinges and strikeplate but security becomes an issue. It's very easy to break in with the hinges on the outside because you simply pop out the hinge pins and the door comes off. But honestly, if someone wanted to break in they could just bust the window with less effort. There are security type hinges (non-removable pin) you could also use.
You would also need to switch the door stop moulding. On newer homes the door stop is actually part of the shaped jamb and cannot be removed. Your door looks older and the door stop is most likely a separate moulding which you can simply remove and nail back the opposite way. You might want to use a utility knife to cut the paint before you pry it off.
All together, with the right tools, this should be about an hour project.