If yoghurt is a variant of yaourt, why is the g pronounced?

The <ğ> used to be pronounced, as a [ɣ]. It has dropped out in Modern Standard Turkish, though it survives in Turkish dialect, and in Greek loanwords from Turkish. So yoğurt used to be [joɣurt], which was transliterated as yoghurt. The /g/ is pronounced in that transliteration, because that’s the default thing to do in languages that don’t have a [ɣ].

I just said that Greek keeps [ɣ] in Turkish loans; so ağa = αγάς, bağlama = μπαγλαμάς. But in the case of yoghurt, the Greek form is γιαούρτι, which corresponds directly to yaourt, and has no <ğ> in sight.

The Triantafyllidis dictionary’s explanation (Λεξικό της κοινής νεοελληνικής) is that the  yaourt variant has dropped the <ğ> earlier than standard Turkish did, either because it was Balkan Turkish, or because the <ğ> was dropped in Aromanian—for which their evidence is Bulgarian yagurt vs Romanian yaurt.

How do you say “forest” in Greek?

What Yossi Aharon said. The word is Ancient, but I think there are some indications it survived rather than being revived, despite being a horrible no good third declension noun.

A word no longer heard for “forest” is the Turkish loan word ρουμάνι < orman.

How do Greek people pronounce Thalia?

Modern Greek:

[ˈθa.ʎi.a]. If it was a truly vernacular name it would end up in two syllables as  [ˈθa.ʎa], but it isn’t.

That’s Thaglia, with the gli pronounced as in Italian (palatal l).

Ancient Greek, which you didn’t ask for:

[ tʰáleːa]