1) If you do not wear the contact lenses, do not mess with the case. Only open or close it when you are going to put the contact lenses in or out. The less you handle the contact lenses and the case, the less likely you are allow microorganisms in.
But, do make sure you always put in fresh solution when you do put your contact lenses into the case. Never re-use old solution.
And make sure you always rinse your case clean with water when you put your contact lenses into your eyes, and then rinse the water off with solution and let the case air dry until you are ready to put your contacts into the case at the end of the day. Doing this helps to keep your case clean.
And a good rule of thumb is to throw away your old case and get a new one whenever you start a new bottle of contact lens solution. Again, just helps to keep things clean and sanitary.
2) Yes, always put new solution into the contact lens case when you put your contact lenses in the case (See above).
3) I'm not sure what you mean by this. If you are staring at the sun with your eyes (sun gazing), that's NOT a good idea at all and it doesn't matter if you are wearing contact lenses or not. Staring at the sun or being outside without protective sunglasses that block UV light can damage your retina and your optic nerve. In severe cases of sun-gazing where people would stare at the sun for hours, you can actually go blind.
If you have any other questions, I am sure the clinic that gave you the contact lenses would be more than happy to help you as well, if you gave them a call on the phone. They want you to be safe, educated, and happy with your contact lenses.