[RPG] How does Liquid Ice work as a material component for Ray of Frost

No.

It specifically mentions that warlock eldritch blasts and invocations are subject to arcane spell failure because unlike most spell-like abilities they have somatic and/or verbal components, albeit simple ones that can be cast while wearing light armor (or medium with Battle Caster feat).

So in that situation, no they couldn't. And since spell-like abilities are in all other circumstances non-somatic/verbal, there's no Still/Silent Spell-Like Ability feat (unlike Empower/Quicken/Maximize). So you'll never be capable of casting them in that situation.

From Complete Arcane, page 7:

All invocations, including eldritch blast, have a somatic component.

[...]

Unlike other spell-like abilities, invocations are subject to arcane spell failure chance.

I would say no, for several reasons:

Ray of Frost on the PFSRD.

1. Ray of Frost does not freeze things, by RAW. There are spells that specifically say that they cause certain elemental effects (for example, Fireball specifically says that it "sets fire to combustibles") but Ray of Frost is not one of them. While it might be able to cool water or make a very small amount of ice, I doubt that Ray of Frost would be able to freeze enough water fast enough to break a lock.

2. Actually making the water stay in the lock is non-trivial. If you have some way of keeping the water from spilling out of the keyholes, then this isn't as much of a problem, but I don't think much water will actually stay in the lock after you've poured it in.

3. From a balance perspective, allowing an easy way to break open nonmagical, traditional locks using two 0-level spells is kind of cheesy, and Knock is 2nd level. I might be comfortable with players using a 1st level spell creatively this way, and definitely if they can creatively use a 2nd level spell for this kind of effect, but I think that using 0-level spells for a 2nd level effect is pushing it. I realize that Knock does more than just open nonmagical locks, but there isn't anything below 2nd level that can open locks, so I feel like that's a good baseline for this effect.

4. I'm not sure this would actually work, at least not in a reasonable amount of time. While you could certainly make a lock stop working by freezing water in it, breaking the lock entirely requires much more pressure than ice can reasonably exert. I find it more likely that the iron or steel that most locks are made of would just force the ice out of the keyholes or into other open areas rather than break. In the related Mythbusters episode, it required liquid nitrogen (which is likely much colder than a 0-level spell), and it still needed an impact to break the lock.

5. This is much more situational, but some doors (especially doors designed for security) don't open just because the locking mechanism is broken. If a door has a deadbolt attached to the lock, then breaking the lock won't let you open the door anyway. And if this trick works in a particular world, then it's very likely that locksmiths will adapt to this technique, and make locks that stay locked after being broken.