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Here's the scenario: a unit of, let's say, assault marines, charges an IG blob with a commissar in it; I issue a challenge, and the IG player declines, so I choose the commissar as the character who declined the challenge. Now, as per the rules, the character who declined the challenge cannot attack in the assault and his Ld. does not count for the squad, but stubborn is still in effect since the rules don't say that any special abilities the character has no longer work. Does this count as a negative modifier and thus the squad is still Ld. 9?

No, a leadership modifier would modify a base stat Ld.

This is simply using a different model's Leadership.
The leadership would be rolled on the lower "leader's" leadership.

AFAIK in the example you cited:

If the IG lost the combat, they would still be stubborn, but would test of Ld 8 (or 7 if no Sergeants remain alive). The Summary Execution rule would still apply too.

The rule states that the model can't attack and their Leadership can't be used. Nothing I have read seems to remove the other special rules.

Commie refuses the challenge and skulks at the back, but that doesn't stop him from capping any waverers...