In regards to the Gothitelle set with no attacks challenge (which I'm not entering at the moment at least since I don't fully feel like testing this out at the moment), I actually had already been running a Gothorita set that satisfies that specification on a team I was running with both the Goths (I wanted to test out the effectiveness of being able to switch between shadow taggers and Gothorita's 60/70/85 defenses with Eviolite are the equivalent of what a Pokemon with 60/129.5/136.5 or 60/114/152 would have after investment depending on whether you run 252 HP/252 Def+ or 252 HP/252 SpDef+), and I imagine Gothitelle can probably run something similar.
The Gothorita version of the set I was running was
252 HP/252 Def/4 SpDef
It's great for stalling out Choice-locked mons, Life Orbers that either don't boost or are slower and can't hurt it enough to beat it before they die, and weak defensive mons without more than two damaging moves. Besides PP stalling against things that can't break it (which having 128 PP and the ability to not do anything for 32 turns while staying in that Rest brings is very good for), it can also provide the team with temporary Screen support or get members in without fear of nasty status moves. I imagine Gothitelle could run a similar set, albeit less bulky (though Gothitelle could use Leftovers or Light Clay in place of Eviolite). The biggest flaw I have found is that there will be at least one turn where Taunt has worn off and Rest hasn't yet, so phazers will have a turn to Roar or Whirlwind you out when you need to recover and Taunters will have a turn to ruin your fun if you let them amongst other things.