PookaMustard: Doesn't the DOSbox executable come with EA DRM on it? So you can't play it without playing it first on an Origin account? Still the problem is easily solved; find any other Dosbox game on your PC, get the executable of that Dosbox game, then put it into your EA Dosbox game, and play forever.
Not that I'm aware of, and while I'm not sure if DOSBox falls under the GPL version 2 or version 3, but if it's under GPL version 3, attaching DRM to the DOSBox executable without getting permission from the DOSBox team would consistute a violation of the DOSBox EULA.
Besides that, as you say, it would be pointless to do so. I know for one thing that the DOSBox provided with Origin's version of Syndicate is also DRM-free. I haven't bothered checking the Origin version of Theme Hospital because I play the GOG version through Galaxy.