1.Two of the spartans have major plasma burns and linda has somethign fractured (forgot)
2.
A) Perhaps she realized something we do not yet understand. We know a man named Dr ackerson was hacking her computer and finding the locations of where the spartans were born and there DNA codes. This suggests ackekerson believed there was a reason some children survived the augmentations and enhancements while most died. Perhaps she took kelly because she had figured out why or part of the reason why and wished to further study on kelly.
B) Perhaps she realized very shortly the covenant would find earth ( as reach was just found and it's considered the door step of earth) so she searched out mendez who had left 27 years prior to train the next group of spartans. She might of taken kelly to defend her incase there was danger where the new spartans were training.
3. They Wish to search planets for forerunner artifacts. On reach Dr hasely found a crystal that could bend space time (eventually got into tarturus and then truths hands). The crystal allowed master chief to leave to earth to warn them of the covie invasion 13 days before he left the destroyed halo. How ever he leaves halo and about 3 of his days later leaves for earth. So he left 16 days earlier then he would of with out it. While he was heading to earth he was also at halo. Some great theorists (mainly opog) believe truth used this crystal so him and the covie fleet would arrive at earth right after regret left. This way he had a long time to do what ever he had to do before tarturus activated halo. Obviously he didn't plan for the arbiter to stop him because he believed the arbiter to be dead.
Now if finding weapons like that aren't enough motive for the covenant we must remember everything the covenant has was based / was forerunner. They wish to find new weapons , New technologies and new advantages against human kind. Unfortunately mankind doesn't have scarabs to dig up artifacts other wise we would of won the war not long after it started.