They are included as Spells Known

The feature Stormborn reads:

... you gain the following spells at the listed sorcerer level. These spells do not count against the number of sorcerer spells you know.

The last sentence there seems to imply that they are included in the list of known spells from your Spellcasting feature, although in this case they don't count against you.

You do not gain any spell slots from this feature.

Balance and Feedback Concerns

In order to be ready for Xanathar's Guide to Everything, the Bonus Spells were removed. There is no way of knowing whether this was for balance reasons or simply due to playtest feedback, but just be aware that the most updated version is the most likely to fit in well with other characters in your game. If no one has a problem with the old iteration, then you can ignore this advice.

One creature is affected by Thunderous Smite, both affected by Booming Blade

The wording on Thunderous Smite is pretty unambiguous:

The first time you hit with a melee weapon attack during this spell's duration, your weapon rings with thunder...

Regardless of the circumstances of Booming Blade, and it being twinned, only one of those melee strikes will be allowed to benefit from Thunderous Smite.

There's no restriction on Booming Blade being used with Thunderous Smite, though, and Booming Blade is perfectly viable to use with the Twinned Spell metamagic, because it satisfies the requirements of only affecting a single target. So this is still a very powerful combo, even if you can't apply the smite damage to both targets.