If you want to take opened contact lenses in your hand luggage, I think it is OK. This is because the contact lenses is sealed in packs, you needn't worry it will be damages in hand luggage. But if your contact lenses are opened, you'd better not put it in hand luggage because the solution in your contacts cases may leak out and cause some hurts to your contact lenses.
Yes, you can. But there are some tips you should take notice. First of all, you must make sure that the contact lenses are soaped in the lotion which may let them being disinfected. Then, you should put them in the right position. If they are put tilted, the contact lenses may be sticked to the side, not in the lotion. Thus, you'd better watch it now and then. In a word, keeping them in the lotion is the first and foremost important thing for you to do.
Airline Company forbids passengers to carry hand luggage with over 100ml liquid. Contact lenses can be carried in luggage on flight. However, you should note if your contact lenses solution is over 100ml. Usually, when you are in travel, you should choose solution as less as possible. Another way you can choose is to buy disposable contact lenses. It is a good way for travel.