Lilting Lullaby

Yes.

When you counter your own spell, it is cleared from the chain.^[425.1.a] "Spell" in a targeting context refers to objects on the chain.^[355.9.a.2] By the time Lilting Lullaby resolves, its target is no longer on the chain and no longer meets the targeting requirements — this is mistargeting.^{[359.3.e.2][359.3.e.9]}

Because all of Lilting Lullaby's first instruction's targets are now invalid, the "Counter a spell" instruction does not execute.^[359.3.e.7]

"Counter a spell" and "Its controller can't play spells this turn" are linked instructions — the second references the controller of the spell from the first. Since the earlier linked instruction was ignored, the later one is ignored as well.^{[359.3.e.14.a]}