Install storm door on the door frame

The closer has a few purposes:

• it closes the door so that you don't have to remember to every time. It will also pull the door closed if it opens accidentally (for example, if the wind catches the door).
• it prevents the door from moving too fast, and usually prevents it from opening past about 90 degrees. (Anecdote: I know someone who didn't have a closer on his storm door. The door opened so that it would catch the prevailing winds -- which are often strong -- in his area, but he never had problems until one time he forgot to close it and it swung open, hit the wall next to the door and knocked out a pane of glass. With a closer, it might not have been able to move fast enough or far enough to do any damage.)
• it provides a second point of attachment to the door frame for the side of the door that opens, which helps to hold the door if the wind catches it.

Does the door close securely, and will you always remember to close it securely? If not, I'd say you need the closer.

You dont state what your problem is. You can install the door to the outside of the door frame. It looks like the top sticks out farther than the sides so you either have to add a board the thickness of the distance to the outside of the side frame. If you need more room for the frame flange at the top you can add a 1x2 board to the top of the door . With the thick door frame the bottom of the screen door may not hit something so you will have to look at that.

Installing a screen door is not an easy project even on a standard door. Especially the first time so if you're doubting yourself you might want to hire someone.