Ok, so I just started the Beginner´s lessons and have a question. I hope this is the right place to ask? So, the letter "ö" is pronounced as "u" in "urge", right? I´m not a native English speaker so that basically sounds as a shwa to me, is that right?

Anyways, I´m from Bulgaria and we do have the word "göl" in Bulgarian and the "ö" is pronounced as "yo" (so it´s pronounced "gyol") - I was wondering, is that a completely wrong pronounciation? Or are there maybe some Turkish dialects that pronounce it that way?

Yes! And llike the "ö" in Göthe or Goethe (German).

does "Bu çok güzel bir arabaydı." mean: "This was a very beautiful/nice car."?

 Correct ! [Doğru!]

another short question: in the word "yaşındayım" - what´s the "ın"? is that the genitive? so "yaşın" -> "of the age"?

In a simpliest way , it is as you guessed " of " , literally ; At the age of

at =  da

age = yaş

ı =  of

n= buffer letter

-----------------------------------

Yaşında

Example =  Ben 36 yaşındayım ----> I am at the age of 36 [ I am 36 years old ]

in one of the exercises there´s the following example: "benim saatim" - shouldn´t it be "benim saatım"? or is this an exception from the rule (or maybe just a typo)?

Arabic loan words are often an exception what comes to vowel harmony. So is saat. If you are not certain check the word in the dictionary. For instance

saat ,-ti
1. hour.
2. time; time of day.
3. clock; watch; timepiece.
4. (electricity, gas, or water) meter; taximeter; speedometer.
5. an hour´s walk; the distance that can be traveled in an hour.

(www.turkishdictionary.net)

In the word "dükkân" - what´s the meaning of the sign above the letter "a"? In the alphabet lesson there was no mention of that sign...does it change the way "a" is pronounced or something else?

That sign is called " Circumflex accent " [inceltme işareti, uzatma işareti].

"^" may stand over the vowels a,i and u.  [in loanwords]

One of it´s function is; [inceltme]

- Standing over the letter "a" which is preceded by g,k or l, the circumflex indicates that a "y" sound to be pronounced between the consonant and the following "a". In the syllable "la"  the "y" sound is fainter than in the syllable "ga" and "ka".

Gâvur ---> Heathen

Kâr ---> Profit

Lâle ---> Tulip