I can help get you started, but I don't know how to everything you want. Well, this is for hacking the actual run box built into Windows. If this is a route you may want to consider.
1. Get Resource Hacker (aka ResHacker). Take the Shell32.dll file from the System32 folder and open it with ResHacker.
2. Expand 'Dialog' then expand the '1003' folder. Click on the icon that looks like a flower and you will see the run dialog box appear.
3. Edit it the way you want. Right click and delete the things you don't want. Move the 'OK' box next to the box you type in (like you have it on your screenshot).
4. To get rid of the title bar, right click in the box area and go to 'Edit Dialog'. Toward the bottom there are a few thing highlighted in yellow. Find the one called 'WS_CAPTION'. Click it to un-yellow it. Say goodbye to the titlebar.
That will get it pretty much looking like you want. Things not fixed yet: 1. The 'OK' box is not a cool red box with an arrow. 2. The run box closes after you run something.
Maybe some other MSFN patron will help you fix those two things. Also, I don't know how to call up the box, either. But once you find out, all you have to do is put it in the '%userprofile%/Start Menu\Programs\Startup' folder. That will make it start when Windows starts.
More thoughts... Windows File Protection probably won't let you replace it. You will have to disable WFP or integrate the new Shell32.dll in to the XP CD before installing XP. If you don't have WFP problems, then you can replace it, but changes won't take effect until you reboot.
I did a quick one. It's not as cool as the one you had in the picture, but this is what I figured out in 5 minutes and I don't have the will power to spend more than 5 minutes on it. I knew enough to get this far, so I thought I would point you in the right direction.