There are 4 foot wide rolls of medium weight brown paper sold specifically for this ; taped down with blue painter's tape is how I would protect my floors. Take no chances and trust no one as far as protecting the floors.

Fireplace doors are made to fit into a specific space. If you find a set that works for you, and measure your fire box carefully, too, I think you could do it either way. But fireplace doors come with hardware to sink them into the fire box opening so I think you could do it after, too. I guess it depends on how costly the tile will be, and if you don't mind fitting the fireplace doors into the tile. But I would do it first, myself.

If we all liked the same things, it would be a truly boring world. Do you mean you like the small smooth pebbles (rather than larger rocks or stones) -- because if so, you can use that look ... but perhaps not as a construction material. If you truly like the "river rock" or any other non-conventional look, then consider using that look to to decorate rather than to build. If you could find a decorative bowl you truly like, you could fill it with pebbles/rocks and place them in any room in your house -- even more than one room. Does your bath get sunlight? Does your yard get sunlight outside the bath window or elsewhere? The reason I ask about sunlight is that there is now a solar powered water pump that could be put in a bowl or birdbath in the yard ... with small smooth creek/river pebbles below and around it to help hold the solar water pump in place for a water feature for your sunlight room or patio or yard. Sometimes stores -- sometimes even the Dollar Tree-- will sell smooth pebbles for planters/pots and you may find some you like well enough to use for decorative purposes. https://www.google.com/search?q=Solar+water+pump+feature+for+fountain+bird+bath&client=firefox-b-1-d&source=lnms&tbm=isch&sa=X&ved=2ahUKEwj4xtjBtqjqAhUej3IEHdRSChEQ_AUoAnoECA8QBA&biw=1366&bih=654 https://www.google.com/search?q=smooth+pebbles+for+planters+or+water+features&client=firefox-b-1-d&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjF17Xzt6jqAhXlhHIEHa03A0kQ_AUoAnoECAsQBA&biw=1366&bih=654 Let us know what you decide -- do post after pics.

Additionally , you don’t have to do picture frame molding if you can get the tile flooring around the fireplace to be flush with the wood floor. You then frame out that section of floor with thinner pieces of your regular wood floor instead of picture frame molding . It will all look more cleaner .