The people who run the HOA can "allege" anything they want. But, until they can prove it in court, you don't have to give them a nickel.
And they wouldn't be obligated to cover repair of your "unit" because it's your "unit" and not part of the common elements.
The HOA's $10,000 deductible has nothing to do with you. The HOA's policy covers damage to common elements. You have no obligation to repair common elements (unless the CC&Rs say so and we don't know that yet).
Again, they will tell you the moon is made of green cheese in the hopes that you buy the story. Frankly, I don't they have any clue as to what they are talking about. They might not even understand the difference between "unit" and common elements.
That isn't what your insurance company did. Your insurance company took the first step in defending you against an allegation of negligence by determining that you weren't negligent and denying the claim.
They're already fighting it by denying the claim. The next step in the fight doesn't occur until you've been served with legal papers. Then the insurance company hires a lawyer to defend you.
I hear that a lot, too. Generally, it's from people who didn't buy the right coverage and don't understand the coverage they did buy.
Careful. There's a difference between "bylaws" and CC&Rs (Covenants, Conditions & Restriction). Make sure you read both. The CC&Rs are what counts because that's a legal contract that you agreed to by virtue of buying the condo.
As JK points out, might be to confirm that the electric was on in the unit. Otherwise I have no idea. Personally, though, I would consider those people my enemy and not give them squat. But that's just me. You can give them a copy if you want to.

