(I JUST WOKE UP, SO PLEASE NOTE THAT WHEN I SAY 'ITEMS' I AM REFERRING TO CUSTOM EQUIPMENT, NOT IN-GAME SMASH ITEMS. MY BAD)
My opinion is kind of a mix of everybody's on here. If you're a TO, try new things, experiment, the game is still new just off the shelf, and nobody quite knows what to expect just yet. I'm personally running Mute City II, Arena Ferox, and Prism Tower at my tournament as the three counterpicks, something you wouldn't even consider in PM. Remember that in Melee, practically where the competitive scene for Smash as a whole started, they ran things like Peach's Castle in tournament settings. Feel free to experiment, but at the same time, keep it realistic, nobody wants a stage with a consistent walk off, or extremely active hazards.
As for items and move swaps, I think move swaps are totally fine. Adjust for your matchup, it's fair enough. If you're going to a tournament, you're basically expected to have everything unlocked and playable, at least for your own main (if you're not sharing your system for the tournament). For items, I think it's a bit different. Should be up to the TO, and personally I'd limit it to one item per-character so as to not make anything too crazy, and I would not recommend using anything that alters stats directly. Just having a character's speed change drastically or their power can completely alter the matchup, and in a much less consistent way than changing a move can. There are a million billion different alterations for stats, and it can be far too much to keep track of (this item increases speed by 20, this one by 22, this one by 23, this one by 24, and so on).
For items, I think there should be a list of a few items that TOs allow, and (again) none that directly alter stats so as to keep the experience consistent so that people don't need to adjust tournament-to-tournament.