There are push notifications for the following events (they are all disabled by default, you turn on the ones you want and register the device you want them on):
Destination reached, under attack, planet lost, skill done, build done, message received
You can only have one "push" device registered with the game at a time. You can change the device simply by logging in with a different device and setting the push notifications to on. The system will them make that the default device. You will not receive push notifications on your device if you are online playing on a different device. We also will not spam you with push notifications. You can set the interval at which you will receive push notices. Minimum amount of time is 10 minutes.
For the travel distance between systems, if you are referring to how long it takes to go from one sector to the next it's about 30 seconds for a probe. The speed can be increased/decreased depending on the ship and the skills a player has. Naturally if you have the ability to jump to a beacon it's basically a single jump to go from system a to system b across the universe. That's something different though.