From the PHB (p. 196):
Critical Hits
When you score a critical hit, you get to roll extra dice
for the attack’s damage against the target. Roll all of the
attack’s damage dice twice and add them together. Then
add any relevant modifiers as normal. To speed up play,
you can roll all the damage dice at once.
For example, if you score a critical hit with a dagger,
roll 2d4 for the damage, rather than 1d4, and then add
your relevant ability modifier. If the attack involves other
damage dice, such as from the rogue’s Sneak Attack
feature, you roll those dice twice as well.
The example given answers your first question perfectly.
For the spider, the secondary damage is from poison, so is not inherently part of the attack. This can be seen by the fact that the damage doesn't depend on the attack roll, but a separate saving throw.
Critical hits represent hitting a vulnerable area. With poison, it doesn't really matter where you are hit.
This is an extremely potent nerf to rogues
As someone who is currently playing an Assassin-subclass rogue, that class is balanced around dealing moderate amounts of damage normally - not having the same utility as Thief-subclass Rogues in that they cannot ignore the restrictions on magic items like a Thief can, but, in exchange, dealing huge first-turn NOVA damage.
A 17th-level Assassin can use the Death Strike feature:
Starting at 17th level, you become a master of instant death. When you attack and hit a creature that is surprised, it must make a Constitution saving throw (DC 8 + your Dexterity modifier + your proficiency bonus). On a failed save, double the damage of your attack against the creature.
This is designed to stack with the Assassinate feature:
Starting at 3rd level, you are at your deadliest when you get the drop on your enemies. You have advantage on attack rolls against any creature that hasn’t taken a turn in the combat yet. In addition, any hit you score against a creature that is surprised is a critical hit.
So that an Assassin rogue could deal what is essentially 4 times as much damage on a surprised target. That is (supposed to be) their ultimate class feature.
By comparison, a Thief-subclass Rogue gets to take 2 turns on the first round of combat with the Thief's Reflexes feature:
When you reach 17th level, you have become adept at laying ambushes and quickly escaping danger. You can take two turns during the first round of any combat. You take your first turn at your normal initiative and your second turn at your initiative minus 10. You can’t use this feature when you are surprised.
Both are designed around the inclusion of Sneak Attack Dice in the dice doubled for a critical hit. However, if you remove Sneak Attack Dice from the effects of the Critical, one of these is much more impacted than the other:
The Assassin Rogue essentially deals 4x weapon damage and only 2x Sneak-Attack damage, and, if your DM is particularly mean-spirited, then they might not even allow the 2x Sneak Attack damage from a failed saving throw for Death Strike. What goes from a spectacular first-turn feature for the Assassin turns into a mediocre feature that can only be used on the first turn of combat. The Thief rogue, however, still gets to perform two full turns on the first round of combat, so it's not nearly as much of a nerf to them.
And the Thief subclass can still use any magic item regardless of race, class, or level restrictions with their Use Magic Device feature:
By 13th level, you have learned enough about the workings of magic that you can improvise the use of items even when they are not intended for you. You ignore all class, race, and level requirements on the use of magic items.
And, if you compare this to fighters, who get spectacular damage regeneration features (Champion), multiple-attack features (Eldritch Knight), and going so far as to ignore falling unconscious and interrupt the current turn when dropping to 0 HP without dying outright (Samurai, from Xanathar's Guide to Everything).
In the end, combined with the effects in this question about removing extra damage for critical hits on spell attacks, it strongly seems like this DM is trying to railroad your party into playing Fighters, Barbarians, and Monks.
Best Answer
It would be very unbalanced, especially in the long-run
I understand some of the logic behind this: some cantrips can critically-hit, and those would deal lots of damage when they do hit despite being cantrips. However, what the DM fails to realize is that a Rogue can also do this much damage or more with a Sneak Attack Critical, and on a hit-die (1D8 for the Rogue) that is equal-to or greater-than the three typical dedicated spellcasters: Wizards (1D6), Sorcerers (1D6), or Warlocks (1D8).
This means that a smart-enough Rogue can deal more damage on a critical, has more health than a dedicated spellcaster, and does so on a higher frequency, since, to get Sneak Attack Damage, they have to be rolling to hit on-Advantage (which is why some Rogues never go anywhere without an ally near them), which improves the likelihood of at least one of the dice rolling a Critical.
A Critical Hit doubles the number of dice rolled for damage, or, for some DMs who would rather simplify it, it doubles the amount of damage rolled on the standard amount of dice.
Let's compare:
Saving Throw Spells
But there are other spells, yes. These force the opponent into performing a Saving Throw and can never critically succeed or critically fail. Many of them, especially at higher levels, still deal damage when the opponent succeeds their Saving Throw, but don't do as-much or don't cause additional effects.
In a sense, for a damage-dealing capability, the removal of the ability of Attack-Spells to critcally-hit makes them much-less preferable than Spells that cause Saving Throws, since Ranged Spell Attacks on-failure never do damage while most of the Saving-Throw spells are still useful when the opponent succeeds their DC.
But wait, your DM is allowing Critical Successes and Failures on Saving Throws! And this is where the balance really gets broken: Most of the spells that used Ranged Spell Attacks are designed to do moderate single-target damage, while the spells that cause Saving Throws either do a lot of damage and/or bestow a crippling effect on one target, or do a sizable amount of damage and/or a strong effect to multiple. Think of the Cleric's Spirit Guardians spell, which deals 3D8 damage on a failed save and half as much on a successful save. Adding critical successes and fails means that any opponent that enters the area or starts their turn there could take as little as 1/4th damage on a critially-successful save to as much as 6D8 damage on a critically-failed save.
Essentially, this ruling makes a great portion of the spells on the Spell lists much less useful, makes Saving Throw Spells much more preferable, and doesn't compensate the Spell-Attack spells for this reduction in viability.