Straight-forward case. You roll 2d3 cold damage instead of 1d3 cold damage.
Not a weapon-like spell:
Any spell that requires an attack roll and deals damage functions as a weapon in certain respects, whether the spell deals normal hit point damage, nonlethal damage, ability damage, or energy drain.
Not on this list is an ability penalty, such as the one ray of enfeeblement applies to the target. Thus it cannot score a critical hit and there is no “damage” to double.
Complete Arcane actually explicitly calls Melf’s acid arrow out by name, saying
Only damage that the spell deals in the round it strikes is increased by a critical hit. For example, if you score a critical hit with Melf’s acid arrow, only damage dealt in the first round of the spell’s duration is doubled.
So only the original 2d4 damage is doubled, to 4d4, and 2d4 damage is dealt each round thereafter.
This appears in two different sources, Planar Handbook and Spell Compendium. It changed pretty significantly from one to the other, so I’ll address each:
Planar Handbook version: belker claws deals 3d4 damage with a touch, and each touched target takes an additional 3d4 damage for a number of rounds thereafter equal to a third of the caster level. The touch attack may also apparently be used repeatedly (exactly how many times is unclear but ultimately not relevant to this question).
For any single attack, the initial 3d4 damage is doubled to 6d4. As with Melf’s acid arrow, the 3d4 damage each round thereafter is unaffected.
When you use the spell to make multiple touch attacks, each touch is a separate attack, a separate roll, and a separate opportunity to roll a critical hit. The initial damage is doubled to 6d4 for any attack that confirms a critical, regardless of how long the spell lasts. Note in the quote above that the rule focuses on “the round [the spell] strikes,” not when it’s cast. Strike multiple times, and roll multiple critical hits, and you double the damage multiple times.
Spell Compendium version: belker claws deals 2d12 damage up-front. It is unclear if “the smoke lasts another round” refers to the smoke on your hands (allowing you to make more attacks) or if it refers to the smoke you are shoving into someone’s lungs (making it a damage-over-time effect).
Either way, though, the answer is clear: if it’s damage-over-time, then only the initial damage is doubled. If it’s multiple attacks, then you can double it to 4d12 every time you roll a critical hit.
Identical situation to Melf’s acid arrow, the initial eldritch blast damage is doubled but the lingering 2d6 fire damage per round thereafter is unaffected.
If you want to say that the 2d6 fire damage per round actually starts the same round as the initial brimstone blast hit (consistent with something being set on fire, but not apparent from the brimstone blast text or mentioned in its example), then that is the same round as the weapon-like spell “strikes,” per the rule about damage over time, and thus would be doubled to 4d6 (still remaining as 2d6 in subsequent rounds).
As with belker claws, separate attacks should roll separate critical hits. But there are questions.
First, if you score a critical hit on one of the later attack rolls, one would expect that doubling the halved damage would just cancel out. Arguably, however, you should use D&D’s peculiar multiplication here, for a \$\times\left(1-\tfrac{1}{2}+1\right) = \times1\tfrac{1}{2}\$ multiplier, which is decidedly odd. Odd, but probably not game-breaking or worth getting worried about. It is worth asking the DM about though if you plan on expecting the 50% more damage there.
The bigger question is what happens if you score a critical hit against the first target. Eldritch chain states that “Each target struck after the first takes half the damage dealt to the first target.” There is no actual wording there that says that you should ignore one-off modifiers to that damage like a critical hit, and since the damage of the subsequent hits is based on the damage of the first, the doubled damage seems like it should pass along to the remaining targets (and be eligible to score their own criticals, increasing their own multiplier and then you get into a whole argument about whether that should be additive or multiplicative with the original critical hit multiplier).
Strict RAW, without wording saying otherwise, it seems like eldritch chain can have an initial critical hit pass that damage on to each subsequent hit. And, again RAW, there isn’t any “memory” of how that damage got that way, so you wouldn’t count its \$\times2\$ multiplier from the critical hit as part of it—it just is the damage it is. You halve that, and in the case of a subsequent critical hit, you then double that halved damage (which again may mean a \$\times1\tfrac{1}{2}\$ multiplier). Definitely talk to your DM before you expecting any of that to work, however.
Personally, I would probably allow it, just because critical hits are rare enough that I wouldn’t be overly concerned about it—not enough to take a player’s toys away, anyway. If you really started optimizing it, though, I might put my foot down.
