Mew usage

My preferred and recommended sets go as:

Shadow Claw/Dragon Tail + Dragon Claw

Shadow Claw/Dragon Tail + Psyshock

These are dodge-friendly, and fun sets to use.

Currently I’m running DT+DC, but would like Psyshock eventually as it outperforms Dragon Claw and has generous timing and is not resisted by most META gym defenders. I simply enjoy Shadow Claw more than DT, and it is faster, but is resisted by the META defenders.

I feel like Mew should use moves with as many neutral matchups as possible since there are normally better options for specialization. Dragon moves are also super effective to the Lati-twins, so in larger groups Mew can be used there to some avail.

For more serious matters, there are better options all around. Mew is more of a fun Pokémon to use than a powerhouse for anything in particular. I wouldn’t worry too much about min/maxing here

Also, the sets I mentioned do at least 89% of Mew’s highest dps:

DT+Psy - 98%
DT+DC - 90%
SC+Psy - 96%
SC+DC - 89%

It's useful as a slot in your dex. Even if Mew's stats were better, the amount of TMs required to get a useful moveset would make it annoying at the least.

My main use for Mew is to enjoy moves that most Pokemon don't get.

I rolled Cut/Dazzling Gleam, and immediately rerolled to Shadow Claw. But I see Dazzling Gleam all the time, and wanted to shoot for Rock Slide or Psyshock instead. After five TMs I got Wild Charge, and I stopped there because it's so good. But I'll eventually try again because I get to use WC on Raikou all the time. Mew is for fun, pure and simple.

Yes.

The research tasks, "Use a super-effective charged move in a gym battle", and "Use seven ... (same thing)". In order for the attacker to get a charged attack off to meet the research objective, the defender has to survive some fast attacks. So if you bring in your maxed out Kyogre against the weakened Golem or Rhydon, you won't get to fire the charged attack off. The key is to balance the foe with something of comparable or slightly higher CP and a three bar charged move to get it off faster. Even the "Seven SE attacks" is pretty easy to accomplish if you set it up right. A gym with 4 motivated pokemon will allow you 12 total battles, and it's not hard to hit the SE charged attack in 7 of those.

I'm working on evolving CP 300 - CP 500 Eevees to get Aqua Tail Vaporeons, Discharge Jolteons, and Flamethrower Flareons with CPs between 900-1500. A couple of old Scythers with Aerial ace and X-Scizzors are good for flying and bug.

Mew fits into this CP range nicely with no or a few power ups, and can have Ancient Power, Dragon Claw, Psyshock, Flame Charge, or Low Sweep, and help meet this research task. The two bars are also good, particularly Dazzling Gleam, Ice Beam, and Wild Charge. I'm hoping for Dragon Claw personally, paired with Charge Beam or Infestation (very high energy gain, not so great DPS), could be good to take down other Dragons, and not much else has it. Dazzling Gleam would also work, as well as covering Fighting and Dark, but is only 2 bars, not 3.

It's kind of like prestiging, but with type optimization mandatory, to fulfill the research goal.

I went for shadow claw / dark pulse.

It makes Mew a second tier psychic counter, but most of all they are very visual and matching animations, so it is very fun to look at.

As tittle says... Mew usage...
Maybe Mew is the best Dex Filler in G1...

You kind of described Mew's use yourself. It's at most B tier for anything you could think of. Relatively low attack stat combined with lack of STAB for most moves means its DPS just isn't up to par in a DPS-centric meta. I would probably pick a gym sweeper moveset like DT/DC and use it for fun in gyms.

At first, the huge move pool made it seem daunting to use TMs for a non-meta-relevant mon. Then I thought about it some more:

(1) Key: I've always got more FTMs than I need but CTMs are precious. Want a fun use which works with the typing and original charge move.

(2) Original charge move = Blizzard. Could be used against the Latis. But fast move = Rock Smash.

(3) 6 out of 14 fast moves are SE against Dragon-Psychic so the probability of not getting one with n FTMs is (7/13)^n. This gives 29% for n = 2 & only 5% for n = 5. Well worth a few FTMs.

In the end it took 2 FTMs to get Snarl-Blizzard. I know it's not an amazing Lati@s counter but I'm going to enjoy using it when there's a large group.

Why are people forcing themselves to use b/c tier pokemon? Especially if you actually care about stats, there is a better pokemon for every case.
Mew is rare, you got one. We will all soon have one. Just put it away.

I had 40 CTMs eating up bag space, so I used ten trying to get Dark Pulse, but I gave up. Mine currently has Frost Breath/Ice Beam, but I’ll probably never use her.

I’m planning to play around with it when I finally evolve my Magikarp, and one of my thoughts was Infestation/Flame Charge. Currently, Mew is the strongest user of Flame Charge, a rather strong 3-bar 70 dmg move. It’s an Eggy killer, kind of ok against most Psychic mons and still fun and quick to use and dodge with. Mew can be the lovely weird filler you’ve been hoping for or the nightmare of a trainer obsessed with DPS and numbers, it’s all up to you. I don’t have it yet and I already love it and calmly look forward to having fun with it :)