No, they don't. As someone who's used firebolt with viking funeral since I obtained the set-up, I have never seen it or any of the bolt nades kill full-health enemies in one shot. They have to be all weakened beforehand for those nades to kill.
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It can happen if you land the grenade close enough to them
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No, I believe all the bolt nades do a static 100 damage (enough to take down shields) regardless of how far targets are from the nade. Firebolts on viking funeral add on around 9 tics of 7 burn damage, making them do a total of 163 damage which is not enough to kill any player by itself.