Lost iPhone, if "Lost Mode" is on would someone still be able to access messages etc? I lost my iPhone, but I believe I have "Find My iPhone" switched on. If I turn "Lost Mode" on, using a different Apple device, can messages etc still be accessed by a thief?
No, notifications like messages won't be displayed in the lock screen.

I have just tested it with iOS 10.3.2 with an iPhone 6s. After disabling 'Lost Mode' you will see the messages your phone received in the meantime.