Anyone wielding two light weapons can attack with one weapon and use a bonus action to make an offhand attack.

The Rogue's having Cunning Action as an option does not prevent them from instead choosing other bonus actions. The restriction only applies to Cunning Action itself - if the Rogue chooses to use Cunning Action, only Dash, Disengage, and Hide can be used. Otherwise they may use any other Bonus Action which is available to them. This distinction is there because the Thief and Arcane Trickster archetypes later expand the options provided by Cunning Action.

The bottom line:

1. The Rogue needs the Dual Wielder feat to use Two Weapon Fighting with a rapier, since the rapier is not a light weapon.
2. The Rogue can use either Cunning Action, or an offhand attack, but not both.
3. Having the Cunning Action class feature does not prevent the Rogue from choosing other bonus actions instead.

The attack must be made as part of the Attack action

I can see your reasoning, but at the very least that goes against what I believe the intent of the wording is. The obvious reading of the rule is that you attack with the Attack action using a light melee weapon, then you can use your bonus action to make another attack with your other light melee weapon.

The rule you're referencing:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand.

I read this to mean that they must both be done at the same time. You have mentioned other ways this could have been worded to achieve this intent that would not provide you with your loophole (Action Surge and Booming Blade, etc) but this is the wording they went with, and I believe their intent was for "and" to mean "both of these things must be done as one action", although it doesn't explicitly say that.

Although not exactly about this, Jeremy Crawford has said the following, which broadly covers my reasoning presented here:

In any piece of writing, context matters. If a rule has multiple sentences, they're meant to be read together. For example, the first sentence of Divine Sense is meant to be read with the rest of the feature's sentences, which explain that first sentence.

Hence when you take the Cast a Spell action instead of the Attack action to make the melee attack with a light weapon, that does not satisfy the "and" part of that clause.

Without clarification on this exact issue from a designer, this is probably the best I'm going to be able to conclude.