When you play Pokemon, do not settle for the worse ability. Don't settle for a bad nature, don't settle for weaker items like Mind Plate and Twistedspoon. Go out of your way and trade for extra Leftovers. Go to the Battle thing and earn Assault Vest, Choice Items, Life Orb, etc. Go big or go home I always say.
Alakazam, use Sub + 3 attacks. Protect is vastly inferior. If you sub on the switch you get a free attack. If they don't switch you scout out what they were planning to do to you by staying in. Death Fodder, etc.
Jynx, Lovely Kiss has 75% accuracy to sleep a Pokemon. One of the better sleep-inducing moves. Frost Breath at first glance seems really good, except Critical hits only do an additional 33% damage as of Gen 6. 80 BP with 90 accuracy? Opt for Ice Beam. You won't need to go through many Calm Mind users with Jynx anyway. Also, Oblivious is neat since you can't get Taunted, but I still prefer Dry Skin. Good to switch in on a Water move for some HP restoration. A lot of people overlook this as well. Nasty Plot is always an option too.
Ice Beam/Psychic/Lovely Kiss/Nasty Plot wouldn't be too bad.
Exeggutor is neat with Harvest. 50% chance to recover Berry at the end of EVERY turn. Sitrus Berry spam galore. Use Sub + 3 attacks or 2 and Calm Mind with Sitrus Berry for some ownage.
Swoobat, use Simple (Hidden Ability). Doubles stat boosts. +2/+2 from Calm Mind. And don't use Rash, I'd choose Modest because 77 Base Sp. Atk is underwhelming and his speed is fine. Modest or Timid for sure. Also, given you're running Mono-Psychic, not everything has to run Psychic STAB. I'd opt for Charge Beam/Shadow Ball/Energy Ball with Air Slash. Roost is optional, I'd advise against it since you're wide open on the physical side anyway. It will be hard setting up too..
Slowbro, don't run that set. Run the Regenerator set with Slack Off/Scald/Flamethrower/Toxic. Flamethrower is optional, but don't use Psychic as mentioned above. Ice Beam or Flamethrower. This is an important Pokemon to get right on Mono-Psychic. He is the prime physical wall.
I honestly don't think these Pokemon will make the cut: Swoobat, Delphox, Gothitelle, Gardevoir. Swoobat's Base Stats are too underwhelming, Delphox's list of weaknesses seals its fate 10 times out of 10, Gothitelle is outclassed at any role. I've tried Gardevoir, its resistances fall flat. It can't switch in on anything, it's weak physically and slow, and just slows your momentum down. Plus it's walled easily unless you depend on Focus Blast's shaky 70% accuracy.
Jynx could do some work player properly, same with Malamar. They're not "good", but they can work.
Slowbro, Alakzam, Jirachi, Metagross, Medicham, Starmie, Azelf, Celebi, Victini, Reuniclus, Exeggutor are what you should be looking at first and foremost.
I also want to point out that Wobbuffet and Mew are NOT Uber. Abuse this!!
EDIT: Remember, NOT EVERYTHING needs Psychic STAB. Azelf should always run it because he has no other STAB and his options in attacking moves are somewhat limited. A Ground-type team might use EQ on 5 Pokes and Earth Power on another because Ground has great coverage. Psychic paired with any 1 other type does not have great coverage.
Good type coverage in 2 moves:
Water & Flying/Normal. Ice & Electric. Ground/Fighting & Rock. Ghost/Dark & Fighting.
There are Pokemon that resist both, but for the most part these are the big type coverage pairings.