I won't double up on the others' corrections. But I think they might have misunderstood what you were trying to say for your age. なんさい means 'how old?' so it's used when asking someone's age, but not when answering. You need to omit the 'なん’ (literally 'what?'). I'm not sure if your なん was a mistyping of 7 (nana) or an accidental inclusion of 'what' (nan).

If it's a mistyping of 7, then you meant to say you are 37, right?

If so, it should be 三じゅうななさいです

Or if you meant that you are 30, it's 三じゅっさいです

All the numbers (not just 三)　can be fully written in kanji too, like 三十七さいです (37 years old) or 三十さいです (30 years old). There is also a kanji for さい, but it's pretty hard: 歳

私のかぞく

こんにちは。私の名前はHollyです。三じゅううんさいです。私のごしゅじんのなまえはMarkです。かれはさんじゅうろくさいです。

私のかぞく

こんにちは。私の名前はHollyです。三じうじゅうなんさいです。私のごしゆじん、なまえ主人の名前はMarkです。かれはさんじうさんじゅうろくさいです。