If you lock out your iphone and put it in lost mode does it still connect to internet?
Putting an iPhone in Lost Mode does not stop you from seeing your phone's location. It may be offline because the battery drained. It's also possible for a person that has the iPhone to power it down in the normal way using the Sleep button and on-screen slider. Regarding being notified of its location if it connects to the internet again, you may find this information helpful: 

From http://support.apple.com/kb/PH2698:


  If Find My iPhone can’t locate a device, the last known location is displayed for up to 24 hours. Select “Notify me when found” to get an email when it comes online.