Insurance in place, is it feasible to insert a clause in a Rental Agreement stating that for example, "Tenants agree that they will bear full responsibility for any damage, whether through negligence, deliberate act or accident, occuring from fire in any fireplace the determination of which shall be concluded forensically through municipal/county/assign-your-jurisdiction fire inspection" or something of the sort?

You, as members of the board, can be sued for negligence if something were to happen.

If something was working when they moved in and it broke (and not from their own negligence, which is in the lease), then I fix it.

The corporate veil is a function of law which usually protects the individuals who run the corporation from personal liability even when they are individually negligent, if the negligence was committed while they were working for the corporation.

Most landlords I speak with regarding Section 8 only take those tenants where the tenant-paid portion is so small that the landlord is willing to forego collecting it; their expectation is that it will be unpaid at some point, so if it is a negligible amount they just don't worry.

Then again, in the short time it can take to flip a house,relatively speaking, appreciation could be negligible in either market.

Simple negligence is not sufficient to open an LLC member to liability.

It may have an almost negligible effect or in some unusual circumstance may have a devastating effect.

Also, is it common for the insurance company to conduct a very thorough investigation to see if the fire is due to landlord's negligence so that they can deny the claim?

If they cause damage intentionally or due to negligence, you can deduct the cost from their security deposit at move-out.