The flange will sit on top of some tile we have down, and we got a wax seal with an extension to reach down past the flange. Otherwise, we have no way to secure the toilet.

Doesn't the flange have a slot for the base screws? Even if it is flush with the floor, you can slip the screws into the flange. If all else fails, drill a hole through the tile and use a floor screw to go into the wood floor and then use the machine end to tighten down the toilet. Many older houses did not have a floor flange.

I must have misinterpreted the question, sorry. Yes, the flange can sit on the tile floor. Ideally it should be flush but most outlet horns will sit down on the flange. If necessary, chip out the tile and drop the flange or use toilet shims.

Closet ring sitting on top of the tile is perfectly OK. Milo