The warlock's ability to regulate (to some degree) the effects of her eldritch blast (like damage) by lowering the blast's effective caster level has no impact on her ability to pick or employ metamagic-like feats related to the eldritch blast.
The eldritch blast's spell level
The warlock's effective caster level of her eldritch blast isn't the same as the eldritch blast's effective spell level. (This overuse of the word level dates to the game's inception, even inspiring a favorite Web comic strip.) The spell level of eldritch blast was changed by the Complete Arcane errata:
Page 7: Eldritch Blast
Second paragraph of the Eldritch Blast ability description: Change “An eldritch blast is the equivalent of a spell whose level is equal to one-half the warlock’s class level (round down), with a minimum spell level of 1st and a maximum of 9th when the warlock reaches 18th level or higher” to “An eldritch blast is the equivalent of a 1st-level spell. If you apply a blast shape or eldritch essence invocation to your eldritch blast (see page 130), your eldritch blast uses the level equivalent of the shape or essence.”
Any other references to eldritch blast being something other than the equivalent of a 1st-level spell should be disregarded. Any other references claiming that eldritch blast is not an invocation should be disregarded. A warlock can use eldritch blast at will. (1)
This means for most purposes except as pointed out by the errata the eldritch blast is effectively a 1st-level spell. Thus, for example, a warlock must possess at least a caster level of 6 (therefore usually being a level 6 warlock) to take the feats Empower Spell-like Ability (eldritch blast) (Monster Manual 303) and Maximize Spell-like Ability (eldritch blast) (Complete Arcane 81), and a warlock must possess at least a caster level of 10 (therefore usually being a level 10 warlock) to take the feat Quicken Spell-like Ability (eldritch blast) (Monster Manual 304).
The eldritch blast's caster level
Because, except for a few situations (like, for example, its lack of components), "a spell-like ability functions just like a spell" (PH 180), this means an eldritch blast can be employed "at a lower caster level than normal, but the caster level you choose must be high enough for you to cast the spell [i.e. employ the blast]…, and all level-dependent features must be based on the same caster level" (PH 171). Thus, a level 7 warlock could opt to employ the eldritch blast as if her effective caster level were, instead, 6, 5, 4, 3, 2, or 1.
For example, to bring this all together, if she wanted to for some reason, a level 7 warlock that possesses the feat Empower Spell-like Ability (eldritch blast) can use the feat yet employ the blast as though her effective caster level were as low as 1.
Your interpretation is correct.
Invocations like Armor of Shadows and Eldritch Sight do not require you to already know the spell, and instead let you cast it for free. If you did have to know the spell beforehand, it would be listed as a prerequisite.
Warlocks can't learn those spells otherwise
It's actually impossible for a single-classed warlock to know those spells, given that Mage Armor and Detect Magic are not on the Warlock spell list (PHB 210)
Best Answer
It’s unclear, unfortunately, whether the “selection” is a thing done once, when you take Quicken Spell-like Ability, or if you must constantly re-select the same choice every time you want to use it—that is, whether or not you have to re-check the spell-like ability’s spell level each time.
If you don’t, then as long as you had CL 10th when you selected eldritch blast (which is necessary to meet the feat requirements anyway), you could select eldritch blast as the spell-like ability in question, and then no matter what changes happen later, you could use quicken eldritch blast.
On the other hand, if the selection is something you have to do again each time you use Quicken Spell-like Ability, then you have an issue. A brimstone blast may still “be” eldritch blast for the purposes of being the spell-like ability you chose originally, but brimstone blast does change the spell level of that spell-like ability to 3rd. Until CL 14th, Quicken Spell-like Ability would not allow you to select a 3rd-level spell-like ability. Therefore, you would have to wait until CL 14th to use it on a brimstone blast.
I feel that the rules text favors the former interpretation, of selecting just once when the feat is taken, but the rules text was also definitely not written with consideration for spell-like abilities that change spell level. That’s not something that spell-like abilities do—except for eldritch blast.