I hear "skirki" instead of what I assume should be more like "skilyki" ("ly" being like the "gl" sound in Italian or "lj" in Serbo-Croatian). Why is this so?
Вам is dative. Вас is genitive and accusative. You usually need to memorize which verbs require which case.
Literally, the sentence means "How many years are there to you?" "To you" → dative.
The usuall age expression is impersonal and uses Dative.
Fortunately, it falls under the pattern that works for a whole range of similar expressions («Мені холодно», «Мені хочеться спати»; even «Йому подобається цей сік», though it is a bit different).
В is sometimes pronounced as w, like in вчора (wchóra) or любов (l'oobów) or Львів (l'veew). I don't know why it isn't w here: it is often w before a consonant or at the end of word. However, років here is not pronounced with a hard в as in Віктор. It sounds more like an f (unvoiced f as in off, NOT the voiced f in of). It might be a regional variation or an irregular form.