Suppose I play Baral, Chief of Compliance with red mana available. The opponent responds with Lightning Bolt killing my Baral.
Can I continue to combo getting a cost reduction from Baral on the stack or does the cost reduction end with the 3 damage from the Bolt?
I believe it does because Baral hasn't resolved yet. In that case, is it a different situation if the opponent targets Baral after it has resolved, for example the following turn?
If you play it right, you can cast one sorcery and a number of instants at a discount.
Two key points before we start:
Lightning Bolt targets a creature or a player. "Creature" means "creature permanent", which means "creature card or token on the battlefield". As such, Baral can only be targeted by Lightning Bolt when it's on the battlefield. In other words, Baral couldn't have been targeted by the Lightning Bolt while it was still on the stack.
If your opponent specifically said they cast Lightning Bolt in response to Baral and proceeded to target Baral, they made a mistake.
If they simply said they were casting Lightning Bolt, they were actually proposing the following shortcut:
You don't have to accept the entire shortcut. For example, let's say one of the spells you wanted to cast is a sorcery. You could let Baral resolve, and step in to cast the sorcery instead of doing step 2. (See [CR 719] for more on shortcuts.) The following would happen instead:
Now, they have priority. Let's say they cast Lightning Bolt at this point. (They don't have to because you shortened their shortcut.) The stack:
At this point, Lightning Bolt hasn't resolved, and thus hasn't dealt its damage yet. That means that instants cast in response to Lightning Bolt will benefit from the discount.
(You could also cast the instants one after the other instead of in response to each other. That would actually be a bit more natural.)
Eventually, after Lightning Bolt resolves and before anything else happens, Baral will be moved to the graveyard and will no longer provide its discount. The continuous effects from static abilities apply at all times that the permanent generating it is on the battlefield or the object generating it is in the appropriate zone. And not a moment longer.[CR 611.3b]
If I play Baral with red mana available, the opponent responds with Lightning Bolt killing my Baral
Impossible. Your opponent must wait for Baral to resolve if they wish to target it with Lightning Bolt, so they can't cast the Lightning Bolt in response to you casting Baral.
can I continue to combo getting a cost reduction from Baral on the stack or I've got a feeling it does because Baral hasn't resolved yet.
Baral static ability only works when Baral is on the battlefield. If Baral hasn't resolved yet, it doesn't provide a discount.
does the cost reduction end with the 3 damage from the Bolt.
Yes. It happens when Baral moves to the graveyard, which happens immediately after the damage is dealt.
This cannot happen the way you describe. You cannot kill Baral, Chief of Compliance, or any creature, with damage while he is still on the stack, creatures on the stack can only be countered, spells like Lightning Bolt can only target creatures on the field. Also you do not get Baral's cost reduction benefit while he's still on the stack, only once he resolves.
Once Baral resolves, you have priority (assuming you didn't somehow flash him in and it's someone else's turn), so you get the first choice of playing a spell, this can be any spell type, so you can choose to cast a sorcery at reduced cost here. Your opponent then has his first chance to cast Lightning Bolt to try and kill Baral, Lightning Bolt will go on the stack like every other spell does, the damage does not happen and Baral does not die as soon as they cast it.
Just like your opponent could respond with lightning bolt to that, you can respond to lightning bolt with a spell of your own, at this point the stack isn't empty so you can only cast instants or cards with flash. The instant you cast can be something to counter the lightning bolt (you are playing blue after all) and save your Baral, or something that will make Baral stronger and let him survive the bolt, or anything else you want.
It's a common shortcut in magic to just resolve spells as they are cast, but the way the stack really works is the last spell put on the stack only resolves once all players have passed priority, all players have had a chance to react to the spell being played and chosen to do nothing.
