Runaway Steam-Kin’s triggered ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered.

If Runaway Steam-Kin has three or more +1/+1 counters on it after you’re done paying for a red spell, its ability doesn’t trigger at all. If it has three or more +1/+1 counters on it as its triggered ability resolves, it doesn’t get a +1/+1 counter.

Runaway Steam-Kin has to be on the battlefield for its ability to trigger. Casting it won’t trigger its own ability.