Is it possible for me to drop my 5k time of 24 mins to 19 by next cross country season?
It's likely possible. You should be able to do that if you're fit and you train right. Talk to your coach about it, as that's the person who knows your current fitness level and abilities, as well as your current training program so will know what to direct you to focus on to build speed, and how to go about it to minimize risk of injury.
It's hard to know. It would help to have a clearer picture of your training and injury history. Doing track in the winter and spring will surely help. You will also need to run during summer break. Without year-round running, I would answer "No." I know of one danger you should be concerned about: Over-training. If you work too hard, especially by doing too many intense workouts, you risk over-training syndrome. This will set you back. In addition, if you try to increase your workout load too rapidly, you increase the risk of injury, which will also set you back. Finally, I suggest you consult with the coach. It's part of the coach's job to discuss things like this. And, the coach knows you better than we do.