Did I just ruin the flange on our toilet?
You have probably damaged the seal between your toilet and the flange. This seal is usually formed by a wax ring that forms closely to the flange and toilet base when the toilet is installed. If you've moved the toilet significantly, you may have opened up a hole in this seal where water can leak when your toilet is draining.

It's sometimes possible to improve the seal by readjusting where you've positioned the toilet. But you probably won't have a good seal unless you fully remove the toilet, scrape out the existing wax ring, and re-seat the toilet using a new wax ring. This is definitely worth doing because a leaky wax ring can drip water into your subfloor for a long time, causing rot and other damage. Plus at the same time you can turn your flange and/or flushometer so you get a better fit between the flushometer and toilet.

You've probably only damaged the wax ring. However, if your toilet remained bolted to the flange while your friend turned it, it's possible you've damaged the seal between the flange and your drain. You should inspect that when you have the toilet up.

By the way, the fitting between your flushometer and toilet is a compression fitting. These are not designed to accommodate an angled installation -- for best results the nut needs to be perpendicular to the incoming pipe. You should try to straighten the joint, and also find a new plastic compression gasket to replace the existing worn one.