As it is the girl who says that she is sure, shouldn't it be "Je suis sûre que ... " instead of "Je suis sûr que ... ?
Also, is "Ce n'est pas grave" (with silent "ne") an accepted alternative for "C'est pas grave" in writing dictation? (I often can't hear all the little words in spoken French but still try to include them in my written responses)
Yes, it should be je suis sûre if a girl is saying it. I believe Cécile already corrected the text.
As to your second question: c'est pas grave is an informal but often used way to say ce n'est pas grave. It is grammatically still considered incorrect even though it is widely used.
I think if you are trying to improve French listening skills with dictation, try to write exactly what you think you hear - If your transcript doesn't match the 'official' one, listen again. Sometimes the transcript will be wrong, sometimes it is just very difficult to hear, which can arise for a number of reasons, including of course just not 'having the ear' yet. To your example - in everyday speech you will hear 'c'est pas grave' or just 'pas grave' (or something in between the 2) very often but you will also still hear some who clearly say 'ce n'est pas grave'. If you are writing informally, go with what you hear. If transforming to more formal writing, you should use formal structures, unless clearly direct quotes.
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