Translation:I only know her last name, not the first name.
Why is it "השם הפרטי" (with a ה in השם) but שם המשפחה (without a ה). Is it because of the שלה?
The difference is that השם הפרטי (literally "the private name") is noun+adjective, whereas שם המשפחה (literally the family name) is a construct state - two nouns.
Yes and no. When it comes to genuinely Hebrew words, some of them use one letter and some use the other. You simply have to memorize them, just as in English you have to know that "s" sound can come from an s (sell) or a c (cell).
However, when it comes to words that are imported from European languages, there is a pattern. They usually use ט to mark a hard "t" sound and a ת to mark a th sound. For instance, "theater" becomes תאטרון.
Most scholars believe that in biblical times the letter ת, without a dot, was pronounced "th". (With a dot it was a "t".) Living in countries where the dominant language didn't have a th sound, Jews started pronouncing it differently. In Ashkenazic countries it became an "s", and in Sephardic countries it became a "t".