How are words with double letters, like şiir or saat, pronounced differently than they would be if they had the ğ (şiğir or sağat)?
No difference at all since ğ would be silent in those positions (between two vowels). In the case of "şiir", you could spell it "şiyir" and it would still be pronounced exactly the same way. Hence you see a lot of people spell the word "değil" as "deyil" and, in chat/sms, as "deil". All 3 would be pronounced the same way.
Saat and şiir are both Arabic in origin and has the letter "ayn" in the middle between the double vowels. But the sound of "ayn" is quite peculiar to Semitic languages and hard to pronounce for speakers of other languages, so they simply dropped it out in Turkish and wrote these words with a double vowel. Eventually these words sound as if they had ğ in them, but I assume that ğ is used only in words of Turkish origins.
"Ayn" is completely silent in Hebrew. So it's rather peculiar to Arabic.
Ğ has rather to do with "ghayn".
istifrağ = إستفراغ ; rağmen = رغماً
True ğ replaces "ghayn" in Turkish. Even in some original Turkish words, I have the feeling ğ is sometimes pronounced like a very soft "ghayn" or maybe my ears just deceive me, because ğ is replaced with "ghayn" when a Turkish word is used in Arabic, e.g. when they say the name of the Turkish President أردوغان
It is with the back vowels (a ı o u) pronounced like that. For the front vowels (e i ö ü) it is very faint but it can never be replaced by "y". It is not the correct Istanbul pronunciation.
No, it is actually fifty-fifty. With the back vowels (a ı o u) it is pronounced more glottal like "q", and with the front vowels (e i ö ü) it is more palatal. But still, there are many exceptions, especially loan words violate this rule. Words like "kâgıt", "kâr" are two of those exceptions. "kâgıt" which means "paper" is sometimes circumflexed but often it is written in the plain form as "kagıt". The second example has always the circumflex since one has to differentiate between "kâr" (profit) and "kar" (snow).
The right pronunciation of "değil" is "diyil".
Source: An elocutionist woman taught me.