Is the audio right ? Is sounds as if she is saying 'my' cousin (dodatai)
(or perhaps that is how דודתה is pronounced ? As if with a final י ?)
Thanks ! (So we wouldn't hear anything analogous to the final syllable of "בבית חלומותי אשר בראש גבעה" ?) where נעמי שמר sings the word 'chalomotai' ?
Great question! The reason חלומותיי ends with an "ai" is that חלומות is plural. Same goes for דודותיי - my aunts - dodotai.
Females can be called "actors" now in America. Actress isn't the only option.
I've been under the impression that in Israel today, people often tend to drop the possessive suffixes (as in דודתה) in favour of using the less formal של, so in this case most people would tend to say דודה שלי. Am I right, or do people still routinely use the possessive suffixes in everyday speech?
Certain nouns still use them regularly. I couldn't tell you which though. Maybe a native speaker can chime in. (They're also used a lot in formal writing).
The only one that EVERYONE uses is אחות. Some other family members and plural books and a few other things are used by some people and they aren't made fun of. (If you say, אני פותח את חנותי, you'll never be able to live it down unless you live in a community of authors and public speakers.) The word אחי is used in slang as "bro", but even that, they also say "אח שלי" a lot for the same use. (Btw, even the word אחיות isn't used in one word possesively. It's EXCLUSIVELY אחות)
Is this actually common to say? Is it more common to just say בת הדודה שלה... ?
They're both equally as correct. Happens to be, most natives just say (wrongly) הבת דודה שלי, putting the article before the phrase and not before the second word
If most natives do that then it's basically no longer incorrect. That's how languages and language change work.