Ok, what is the purpose of this item? I get Soul Stones are for equipment upgrade, but have no idea what to do with Immortalite Stones. I mean I know it can give 1000 shards, but is there other purpose for them?

Immortalite is used to upgrade items to +6 and higher. Save your immortalite until you're very sure you have an item you'll want to use long-term - you won't find much immortalite until the later levels, and it's barely enough to fully upgrade a single item.

https://deathsgambit.wiki.fextralife.com/Upgrades

Aye, it's for upgrades past +5. In a full clear (picking all the loot, opening chests + killing all the bosses on heroic) you can get about 40 immortalites. I don't know the exact number, it changed slightly between patches because some heroic bosses didn't drop them the first time you beat them, which was a bug.

I don't know the exact scaling for how stats affect damage, but I think upgrading late-game helmet, boots, and cape might be a better use of immortalite than upgrading weapons. Those stat bonuses on the armor and cape add up to some large boosts.

I'm thinking of upgrading Consumed Knight Helm, Consumed Knight Boots, Cloak of Darkness, and Scythe of Reaping to +7 instead of taking one piece of gear all the way to +10 (Finesse build, obviously).

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Finesse:

Consumed Knight Helm
Consumed Knight Boots
Cloak of Darkness

Strength:

Drake Knight Helm
Drake Knight Boots
Ascended's Cape

Intelligence:

Void Helm
Void Boots
Void Shroud

I've farmed heroic bosses to get drake boots +5 i think, drake helmet +8 and ascended cape +8 and all 5 str weapons +9 and kept upgrading from there, although very slowly since I don't get many immortalites per run.

+9 > +10 on weapon gives nice boost in damage, but overall it's better imo to upgrade helmet/cape/boots than the weapon alone. Especially the armour which gives 2 stat bonuses at the same time, like str+vit etc. I don't have armour +10, so don't know if it is just +1 to stats, or better boost at the last upgrade step.