日本人には、見知らぬ人と会話を始めてまだくつろいだ気分にならないうちに、相手の、年齢や地位、既婚か未婚かなどの個人的な事柄を知りたがる傾向がある。
There is a tendency for Japanese to want to know a certain amount of personal information about someone such as age, position and whether they are married or not, before they feel comfortable talking with a stranger.
In this example, what does the うち in ならないうちに mean?
On a side note, I sometimes notice that Japanese punctuation differs from English usage. Why would the writer put a comma after 相手の?
うち means "inside," "between," "while." In this example, it is about the temporal span. As まだ suggests, it is assumed that at some temporal point, the person will become relaxed (くつろいだ気分になる). Until then, there is a time span, and that time span corresponds to くつろいだ気分にならないうち "within the temporal span where one has not become relaxed."
About the punctuation in Japanese, it is true that a standard has not yet settled, and where to put them largely depends on the person, but in this case, the writer put the punctuation to disambiguate the sentence. Without it, there are at least two possibilities as for the modifee of 相手の. That is 相手の年齢や地位 or 相手の...事柄. By putting the punctuation, you can tell that 年齢や地位...事柄 is a unit to which 相手の attaches and modify, which would mean that 相手の is modifying 事柄 rather than 年齢や地位.
In this case, Verb＋ないうちに means something like: while Verb hasn't happened yet...
In this case, ならないうちに also goes with the context; namely, the verb that precedes it (くつろぐ・to feel at home with someone.)
So it becomes: while a Japanese person hasn't had a chance to feel at home yet with a stranger... or more naturally put: before a Japanese person feels at home with a stranger....
For your second question, I believe the writer is partly using the 、 here to break up the sentence in order to indicate the second major subject of the sentence (相手) in this case, as well as to indicate that 相手の is not modifying the following noun, as @Hyperworm suggests in the comments.
