I'm not a Rhodes Scholar, but I am an American student from a rural state school who's been accepted to pursue a DPhil in the sciences at Oxford, with full funding.
Regarding your last question, whether your Fulbright ETA would be any boon, you've already answered the question yourself! Unless I misunderstood, you said at the beginning of your post that you have a compelling narrative from your undergraduate to your Fulbright ETA and now to your professional career and desire to pursue further education. If your Fulbright gave you first-hand experiences which are now contributing to your desire to continue your studies, definitely describe that and it'll be a huge boon. But I don't think that name-dropping Fulbright on its own does you any favors or any disservice, unless you can connect it to the motivations underlying your current application in a meaningful way.
Is it bad to have been out of school? No, not necessary. My anecdote along similar lines is that I failed most my classes and dropped out after undergraduate freshman year. I took a year off before returning to university. In a vacuum, neither being out of school as you've been nor being a former drop-out as I am looks good. But applications are not reviewed in a vacuum. Your application should tell a story. I connected my experiences into a cohesive, compelling narrative when submitting my application materials and (with a bit of serendipity no doubt) ended up getting fully-funded offers from both Ivy League schools and Oxford. My experiences are part of my narrative, and I wouldn't be the same candidate without them. In that way, my nontraditional path certainly hasn't hurt me and yours won't necessarily hurt you either.
My final piece of advice is... from an academic perspective, why Oxford? You may want further education for your professional career, but that does not answer the question of why you would like to study at this particular institution. As a prospective student, Oxford (and indeed any university) will be looking for an argument as to you need to study there specifically, why their program is uniquely equipped to address your academic interests. The answer is almost inextricably linked to your academic interests, which is why it is important to decide how you want to specialize sooner than later.