I just did this on my Axiom 49, so it should be similar to set up on yours.
First, switch on your keyboard, load up Reaper and verify that it's being recognized and is active in your MIDI settings. From your picture, I can tell you already did this. (Personally, I'd recommend disabling the Directlink options, since Reaper is not supported with it as far as I know.) Then exit out of your Preferences page.
Now for the important part: from the normal Reaper screen, press the ? key. This will open up your Actions window. In the Filter area in the upper left corner, type in "transport play", without the quotes. A little way down the list, you should see a few options such as "Transport: Play" and "Transport: Play/pause". These 2 are the ones you're most likely to want to use. I set mine to use the Play/pause function, since there's no Pause key on the Axiom.
So click the option you want to select it, then down in the lower left, hit the Add button. It'll prompt you to hit the button you want, so hit Play on your Axiom. It'll automatically recognize the input and you'll see it displayed in the Shortcut: box.
One last thing to do before you're ready to save the action- in the MIDI CC: menu, I had to select the "Relative 1" option to make the button toggle. When it was set to "Absolute", it would only play when I held the Play button down on the Axiom. Anyway, I don't really understand that section too well, but that's what worked for me.
Just repeat this process with your Stop, Record, Loop, etc buttons on the Axiom and you'll be controlling Reaper from your keyboard in no time.
I hope this helps. I realize it's a while after you asked the question nut maybe it'll be helpful to others as well.