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.

Following the discussion here, where a weapon's extra damage also counts as weapon damage dice, then

You can choose either 1d8 slashing or 1d6 fire extra damage for your Brutal Criticals.

Crawford supports this idea with the tweet:

Savage Attacks: add 1 of a weapon's dice to a crit. again. Frost brand is a wpn. w/ 1d6 cold as 1 of its dice

For Frost Brand, which has a very similar wording to Flame Tongue, you can add the frost damage dice to the Orc's Savage Attacks, not only the sword's damage. Therefore, we can extrapolate and assume Flame Tongue behaves similarly with the Barbarian's Brutal Critical.