1) Strictly speaking, if the user does not allow using CoreLocation or, if permission were given but CoreLocation is not available, there is no other fallback to get the user's location as a latitude, longitude pair. Using the locale may not work in all of the cases. First, it will just give you an approximation which may be way too far from reality. However, it's up to you to judge whether this approximation is ok for your app. Moreover, there are some users using a locale different from the one related to their country. And, you have no guarantee that people travelling abroad will adjust the date/time every time.
2) Core Location caches by default the last location retrieved from the GPS unit. However, for people travelling abroad this cached location will be certainly wrong (even for people travelling just a few miles away), and in general, you should discard the cached value and localize again the user each time. I understand that for people travelling within their country this will not (usually) create huge problems. But it just depends on the country: travelling within Italy will not change the time, but travelling across the US may change the time up to 3 hours.
3) I would suggest to try using Core Location and, in case of problems, simply ask the user to input his/her local time or location (the city should be enough for your purpose). If you choose to ask for the user's location, then you can get the corresponding latitude/longitude pair but this will require a working network connection.