1. A group IS ... (Not important who is in the group). A group is a singular, countable noun.
3. Both wrong. She went home half an hour ago. Given, finished time requires simple past UNLESS it relates to another action in the past. e.g. She had gone home before the others arrived.
4. She has sold... Indefinite (no time) past. Done, but we don't know (or care) when.
5 Ditto
6. Have you ever ...? Indefinite past. We don't know when.
7. Mmmm, this one is elementary ! Do you .... ALWAYS requires the infinitive form. e.g. Do you live..., Do you like..., Do you do... Do you have...?
I could write a whole grammar book on how to differentiate the 6 modal verbs and their MANY uses. I suggest you buy one. Good luck :)