What are the most popular Pokemon of this type seen in OU?
Mega Metagross is super popular, and is actually being suspected atm. Latios and Latias have also been staples for more offensively oriented teams, as they are great offensive defoggers. However, their speed tier got a little bit worse in ORAS, because there are a bunch of new base 110 speed megas such as Mega Metagross, Mega Gallade, and Mega Diancie. Starmie is a great offensive spinner, and with Analytic, it can actually hit pretty hard. Starmie is great coverage and can force lots of switches, giving it many opportunities to spin away hazards. Mega Gardevoir was also super common after the Aegislash ban, and during the late XY era it was literally everywhere. Mega Gallade is also seen in OU a lot. Celebi has been rising in usage too. It can run baton pass sets with either SD or Nasty Plot, making it a guessing game of which move it is carrying, or which mon it'll pass to.
What other Pokemon of this type also claim some sort of viability in OU?
Even though cress is only RU, it is probably the most reliable answer to Landorus-I on stall teams, but even then, it's vulnerable to being worn down because of moonlight's low PP, and it can also be pursuit trapped. AV Slowking can handle Mega Metagross pretty well, while also being able to beat Keldeo, a top tier threat.
What Pokemon of this type fill a niche in OU which is sometimes worth considering in a team?
Victini is sometimes considered on stall teams, as it can handle most fairies rather effectively and isn't too passive, as v-create hits like a nuke. Jirachi is also in the same situation as Victini; it handles fairies especially well, and can also provide wish, healing wish (although not commonly seen on stall teams), and SR support.
What sorts of Pokemon does this type collectively threaten?
Mainly fighting-types. Most fighting-types have trouble dealing with bulkier psychic-type Pokemon. Keldeo has a hard time against Latias, although it can be worn down by scald burns and entry hazards, Gallade and the rare Medicham have trouble against normal Slowbro and mega slowbro. Starmie is a solid offensive check to lots of mons in the tier, such as Heatran, LO Breloom, Keldeo, the list goes on.
What sorts of Pokemon threaten many of the members of this type?Bisharp is rather annoying for Psychic types to face. Sucker Punch can take out faster mons like Latios, Latias, and Starmie, while knock off is in general annoying for any Pokemon that prefers having it's item. It also can run pursuit, creating mind games against Pokemon such as the Lati twins. Heatran is also a pain in the butt to deal with, as it resists psychic-type moves, and can use toxic, putting your mons on a timer.
What Pokemon of this type are very potent offensive threats?
Victini is actually quite underrated, it hits like a truck with CB v-create. It can also run a multitude of sets such as scarf, special, or subPUP. Special sets with glaciate are nice to catch landorus-t, a common check to it, on the switch. Mega Metagross is also a great offensive psychic-type mon. It has great coverage with meteor mash, zen headbutt, ice punch, and hammer arm / grass knot, while also being insanely bulky, with a great speed tier too. Mega Gardevoir is an absolute nuke with pixilate hyper voice, and beats mega sableye, which is really common atm.
What Pokemon of this type are very sturdy defensive threats?
Slowbro is really bulky, and doesn't really require too much prediction, as it has regenerator, giving it free health every time it switches out. Mega Slowbro is even bulkier, and can run a "cro" set, or a double dancer set with iron defense and calm mind, making it hard to take down. Latias mega is also quite bulky, and because it holds a mega stone, knock off doesn't do a lot of damage to it, while it can set up CM's and make use of boosts accumulated with stored power. It also has reliable recovery in roost.
What Pokemon of this type increased in viability in the transition to the ORAS meta?
Mega Metagross is really obvious, it was basically in obscurity during XY, and ORAS granted it a mega forme, with solid bulk, speed, offensive stats, and tough claws. Latias also got stored power, which was illegal before, and this is one of the main reasons, why CM Mega Latias is used. I guess Mega Gardevoir too because sableye and stall in general was kind of revived, but it also gained a new offensive check which is mega metagross, although the suspect test is still going on; if mega metagross gets banned, it'll get a lot better.
Have any Pokemon of this type benefited from access to new move tutors, abilities, or legal move combinations?
I think there's only mega latias + stored power, can't really think of anything else.