I'm using the 1st gen Moto360 with an S7 Edge and it does give out any notification that happens on your phone. HOWEVER, two things to note:
1) Not all notifications will light up the screen, only priority ones. Some apps (Whatsapp, Facebook's Messenger, Messaging apps) already set themselves as priority and thus light up the watch when you get a notification, but for those that don't, you can manually set them up as such, but it'd have to be in a per-app basis. You do get the vibration for all, though.
2) This only works for actual notifications. Alarms and timers on your phone don't give out a notification, just an action in your phone generated by the Clock app, so these won't do anything on your watch. You can set up timers in your watch that won't do anything on the phone, and if you give access to it, calendar reminders do notify you on both devices.
As for the leather strap, I find mine very comfortable but it's leather. You have to take care of it. Although the watch is waterproof to a certain extent, getting the strap wet and not caring for it properly, say, after a sweaty workout, will cause the strap to stink quite quickly.