I'm playing hell mode as a monk, and playing it in a monkly manner seems utterly futile. If I enter melee range, I get SWARMED, monsters seemingly just teleport all around me, and then they kill me in two or three hits. Dungeon level 7 has been particularly bad, with goats that charge and their blows hit and stunlock me more often than not.

So far I've survived mainly by relying heavily on fireball and chain lightning and lots of potion drinking, but I'd be so much better off playing this way a sorcerer.

I'm at level 31, with these stats:
103 strength
97 magic
67 dex
82 vitality
174 life
164 mana
151 AC
To-hit 118%
49-66 damage
MAX resist magic
MAX resist fire
50% resist lightning

My gear:
Doppleganger's Staff of Fireball
Blessed plate
Veil of Steel
Diamond amulet
Garnet ring of giants
Serpent's ring of sorcery

Can I do better and not just feel like I'm playing a second-rate sorcerer?

The difficulty level "Hell" grants monsters a +30 level bonus, increased health, and increased damage output (as well as other benefits; see Jarulf's Guide for details). While it is possible for a poorly equipped character at level 30 to defeat dungeon level 16 on Hell difficulty if played carefully (and this was once considered a rite of passage, known as the "3@30" for getting 3 dots at the lowest character level where this was achievable in standard Diablo), it is quite possible that a level 31 character is not ready to face dungeon level 7 on Hell difficulty if you are not very careful. Consider playing a lower difficulty or staying in the church levels for now.

You can only be stunned if you take damage exceeding your character level. If you block an attack, you will take no damage, so it will not stun you. The post above gives some advice on improving your blocking. Try to minimize the number of monsters facing you at once, since you cannot block a new attack if you are still busy blocking the old one. This may mean moving very slowly. Advance one tile. Wait several seconds until you are sure nothing has noticed you. Advance one more. Repeat. Yes, this is very slow, but it does let you survive in areas that are otherwise too dangerous for you.

If you have not yet, consider reading Jarulf's Guide for the sections on combat, so you can understand and model how well your character will or will not do against various enemies. As the above poster mentioned, your to-hit and AC could be notably better for what you are trying to do.

If I must restart the game so I can pump up the dex and get better gear, would it make sense to hold onto the staff for now, since more monsters = more XP and drops? I believe I can basically tear through the legions of normal mode hell, reinforcements or none. I can definitely handle hell mode's early dungeon levels.

Also, what's the ideal way to farm for loot and gold? Hell runs? Crypt runs? So far I haven't done any kind of farming, I've just made use of whatever the RNG gave me.

Would it be worth holding onto the jewelry I've got as backup against spellcasters? Resistance seems like it would be more important than tohit.

And I know plate isn't the best for a monk, but that 65% AC bonus beats anything else I've seen yet. It's what made me give up Arkaine's Valor.

I can recommend farming in hell, in particular the bosses because they always drop something. You can do it on normal difficulty, the loot will be the same as on hell difficulty. As for gold, just sell items you don't need (plates are usually expensive) and you won't have any problems with gold.

You are right, the extra resistances are useful and should not be neglected. I'm not sure whether cloned monsters can drop items though.
The best possible equipment is not a necessity, as long as you feel comfortable with your items then go ahead, I'm just pointing out weaknesses that will become noticable in hell difficulty.

I won!

Final stats:
Level 41
174 strength
100 magic
185 dex
115 vitality
237 life
197 mana
186 AC
To-hit 222%
119-155 damage
50% resist magic
MAX resist fire
50% resist lightning

Gear:
King's staff of firewall
Saintly plate
Veil of Steel
Drake's amulet of perfection
Ring of the zodiac
Ruby ring of harmony
*Jade ring of the stars (used instead of ruby ring to get through levels with magic and lightning casters)

I never found a decent monk-grade armor, or any staves of haste.

In the end, the last level was still a nightmare thanks to those fireball-spamming advocates. Even with max fire resistance, getting hit by four balls at once can mean instant death, and with clusters of advocates all over the place, this happened a lot. I had to follow that old Sierra mantra - save early, save often! Stone Curse was my friend, but to stand any chance against big clusters I'd have to freeze half the group, chug potions, heal, rush in, heal some more, and kill one or two before retreating, possibly back to town for more potions. And I still got killed pretty often. Eventually the pack would be thinned out enough to not be near-instant death and I could stone curse and shatter the remainders.

On occasion I'd encounter just one or two in visual range, and then I'd actually use a crappy little bow to plink away at them, healing as necessary. Better that than get in close and face a barrage of fireballs from whatever allies lurked just a bit farther off the edge of the screen.

Succubi on levels 15 were pretty bad too, but at least they pursue you, so I could gradually thin out packs of them by luring a few away, stone cursing them once out of the throng's range, and killing them. The Lazarus encounter was a pain; I had to do some serious finger gymnastics to stone curse him and a succubus, heal, drink a potion, stone curse the other succubus, and then kill them, re-cursing as necessary, without waking up the horde of regular succubi to the southwest.

The blood knights were jokes. I struggled more against spitters and red storms in the levels above. Still, when combined with advocates or succubi, they turn a bad situation into a slightly worse situation. So I'd let them pursue and then kill them out of range from everyone else, so I could return and deal with the actually dangerous enemies without these walking metal walls getting in my way.

As for Diablo himself, once I lured him away from the insane number of advocates inside his inner cloister, he went down like a beetle under a steel-heeled boot. I didn't even need to heal once.