Only when the caster is below 5th level

The errata for the PHB has clarified the restriction further from the original printing:

To be eligible for Twinned Spell, a spell must be incapable of targeting more than one creature at the spell’s current level.

By default, eldritch blast does not have a range of self and is capable of targeting only one creature. It is thus eligible to be twinned.

However, the spell becomes capable of targeting more creatures once the caster reaches level 5:

The spell creates more than one beam when you reach higher levels: two beams at 5th level, three beams at 11th level, and four beams at 17th level. You can direct the beams at the same target or at different ones. Make a separate attack roll for each beam.

Thus, once the caster reaches 5th level and above, they can no longer twin eldritch blast.

Note that eldritch blast (and all other similar cantrips) scale with character level not class level as confirmed (unofficially) by Jeremy Crawford on Twitter. (See Do Cantrips use your character level or class level? for more discussion about that)

No, you do not need to know fireball - the language of "you can cast" enables you to cast it through this feature regardless of whether or not you know the spell. This invocation uses the same language as the ones that allow you to cast spells the warlock usually cannot learn, such as Dreadful Word or Ascendant Step.

To supplement this answer, note that it specifically has a spell known prerequisite of only eldritch blast, not eldritch blast and fireball. It would likely state the latter if it required that you knew fireball in order to make use of it.