You can edit the Summon Sea Elemental ability (or any summon ability) to summon Doomguards instead. The ability shouldn't really matter unless you want different mechanics. For example, you could base the ability off of Summon Serpent Ward if you wanted to summon the Doomguard at a target point instead of instantly.
Anyway, I still don't fully understand your problem as you haven't explained what you want to happen:
"The issue I'm having is when I summon the Doomguard the Warlock dies just as intended, but it also summons the spiders to, and aside from that the "Spawn Spiders" icon isn't even present."
From the sound of it everything is working exactly as it should. The Warlock dies which summons the Doomguard, and the Spawn Spiders ability goes off because the Warlock died.
Are you saying that you DON'T want the spiders to spawn if the Warlock dies from summoning a Doomguard? If that's the case then an easy fix is to remove the Spawn Spider ability from the Warlock right before you kill it.
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Actions:
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Remove Spawn Spiders from Summoning unit
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Kill Summoning unit
Also, some abilities don't have Icons for whatever reason. An easy fix is to give the Warlock an extra passive ability that does nothing, for example Storm Hammers, and use that as the Icon.
Additionally, you can hide ANY ability Icon by setting it's
Art - Button Position to X:
0, Y:
-11.
Note that you'll have to hold the SHIFT key when opening the editor field in order to type in negative values. You also have to make sure that this feature is enabled in your World Editor as well.
To do this, go to File/Preferences/General and enable:
Allow negative real values in the Object Editor.
The SHIFT key is also used for exceeding editor limits, so if a field ever prevents you from typing say "100", you can usually Shift-click the field to bypass this limit and type whatever value you want.
This is how abilities like a Negative Armor aura are created. You edit Devotion Aura so that it's Targets Allowed are set to Enemies instead of Allies and change it's Bonus Armor to negative values.