I want to play a neutral/evil party with characters like Korgan, Dorn Il-Khan, and so on (not sure who else exactly, will plan the actual team later), which obviously excludes any paladin from it.

So I'm thinking to make the main character a fighter/thief wielding two handed swords (pretty much play them as a fighter who just happens to take care of traps and locks) and at sufficiently high level, take the Use Any Item and give them Carsomyr to wield. Will that be possible?

According to what I read on internet it is possible. here.

Yes. Use Any Item is exactly what it says.

From what I have read, that's not quite true in the Enhanced Edition; it apparently doesn't let you use items that are restricted to specific NPCs.

Also, be aware that being able to equip a weapon doesn't actually get you proficiency with that weapon (so a single-class thief may be able to equip weapons she can't ever become proficient in), and that backstabbing is only possible with weapons only a single-class thief can equip, or with magically created weapons (like the ones you get when you are shapechanged into an Iron Golem, which are strong enough to one hit kill the SoA final boss with a backstab).

I guess you mean items like Keldorn's armor for example, not class-restriced items (like Carsomyr) ?

As for the proficiency, I know, that's why if I go through with it, I plan to make my character a fighter/thief, so that they can get it.

Although I'm not sure about that now, thinking of party make up (not just with power/combat, but also party interactions, banter etc. in mind) - the current plan is:
Dorn Il-Khan
Hexxat (for interactions etc, technically I wouldn't need another thief)
Neera (need a mage, Edwin is annoying, and Jan doesn't get Necromancy spells; also romance, sue me ;) )
Korgan (he apparently has somewhat interesting banter with Dorn)
And Imoen, once I get her (sure, she doesn't fit the neutral/evil party, but I'm just not leaving her behind, don't have to play a good character to have them care about someone :P ).

If I go with fighter/thief PC then, that leaves me without a cleric (thus healer).

So, my options I think are:
Drop Hexxat for some cleric (Viconia? Anomen is a jerk, and has crap Wisdom, which for this party wouldn't get boosted, as he'd fail to get his knighthood).
Or drop the fighter/thief idea (so no Carsomyr) and make PC a cleric or some combo of one.

Suggestions? Does Hexxat have any good banter and/or quests? Is it possible to play through the game without a healer?

Also for Dorn's quest, is it true what I've read, that if you backstab both guys at the end (I'm trying to keep is relatively spoiler free), he will lose his class powers?

Yes, that's what I am referring to.

I know from experience, that if in the end of his BG2:SoA quest you choose to choose to destroy/betray both demons he will become a "Fallen Blackguard." So say goodbye to his Divine Spells. If I'm remembering correctly, you lose all the Blackguard benefits, but I know it doesn't mess up his ToB quest any.

If you really want to be wielding Carsomyr, keep your choice of NPCs, and still have a cleric, then you can consider a Cleric/Thief multiclass.

You'll have to be a half-orc or gnome, your armor choice will be restricted by your desire to use certain thief abilities, and you'll be unable to backstab except with very particular weapons, but you can get UAI and cast healing spells.

I recall that after getting UAI, you'll be able to start adding proficiency points to any weapon slot, at least in non-EE. I don't know if EE changes this. So you'll need another proficiency point AFTER you get UAI, but you'll eventually be able to swing Carsomyr without that penalty.

From a RP perspective it actually is quite interesting for an evil PC to drag Imoen along. Very much an inversion of a good PC bringing along Sarevok. Third option is to rely on items and resting, which is tricky but doable. Hexxet has built-in regeneration, meaning you'll need three other regeneration items to cover Dorn, Korgan, and your PC. If you raise your pickpocket skill high enough you can pickpocket a Ring of Regeneration from Ribald (not sure on how much you need) as well as the Ring of Gaxx from Kangaxx's first form (at which point you can run away and come back to do the fight later). Then head off to Spellhold to get another ring of regeneration and you've got your front-lines covered. Potions are plentiful in Baldur's Gate 2 and can easily tide you over.

This is still a trickier option, but it is entirely possible to play through without a dedicated healer if you don't want them.

I'll probably go with a Fighter/Cleric PC or so (I'm not a fan of pure clerics), but trying to decide on alignment, between Chaotic Neutral or some Evil (probably Neutral Evil).

In BG2EE, Is temple position given to the player based on alignment in the same way it was in normal version?
(as far as I recall, all good clerics got Lathander, all neutrals got Helm and all evil ones got Talos)
I'm asking, since CN doesn't seem to fit Helm all that much for me, and might actually get kicked out, depending on my decisions.

I vaguely recall something about CN cleric getting the temple of Talos, but maybe that was in some mod's readme/website (come to think of it, my idea of the character might get him kicked out of Talos' temple too, so might as well go with Helm and see what happens).

A couple other options:

1. Give somebody Vhailor's Helm and a Rod of Resurrection. Once per day (or in other words, once per time between rests), you can have that character use the helm and have the character's clone use the rod to heal everyone. Later on, you can do this with the Project Image or Simulacrum spell.

2. Heal with arcane magic. Limited Wish can, with enough Wisdom (11 apparently), heal about 25 HP to the party. Later on, once you get HLAs, you can use Summon Planetar; the Planetar can use some nice spells including Heal and Raise Dead. There's also the luck based approach of using WIsh; it doesn't always offer healing, but when it does, it's a full heal to the entire party. (I note that both the Resurrection and Restoration options will heal the party. even characters who aren't dead or drained.)

You can also just use potions for healing. In BG2 in particular there are more than enough Extra Healing Potions to last a lifetime even for a party of 6. Once you get a regenerative item or two you don't even really need potions anymore except in combat.