Yes. Activated (and triggered) Abilities are independent of their sources. The typical analogy is of a hand grenade: If a soldier throws a grenade, and then is shot, the grenade will still explode.
As long as you activate the seal of fire's ability before the enchantment is destroyed (such as, in response to the effect that would destroy it), the effect still triggers (and does the two damage). In addition, the Elvish lyrist's effect would be countered upon resolution (because Seal of Fire is no longer a valid target), though this is of little importance, because the elf has already been sacrificed.
This works because abilities do not resolve instantly once activated. Instead, they are put on the Stack. When an item (be it an ability, trigger, spell being cast, or most anything else) goes onto the stack, each player gets an opportunity (called priority) to respond to that item.
In this case, Elvish Lyrist's ability is activated and placed onto the stack. You (along with any other players in the game) will receive priority in turn. While you have priority, you may activate Seal of Fire's ability in response. It goes onto the stack (above Elvish Lyrist's ability). Once each player has passed priority, items on the stack resolve in the reverse of the order they were placed there (ie, First in, Last Out). Additionally, each player will receive priority after each item on the stack resolves; so a player could choose to cast a spell after seal of fire's ability deals damage, but before Elvish Lyrist's ability resolves.
Yes you can sacrifice the Seal of Fire to do 2 points of damage.
When the Lyrist activates its ability, that ability doesn't happen right away, it's put on the stack. While on the stack, you have the chance to respond, and so you activate Seal of Fire, doing 2 damage.
Then Lyrist's ability tries to resolve, but since the enchantment isn't around anymore, it fizzles, but the costs stay paid (mana and sacrifice).