Is it possible to automatically launch a rocket (with satellite)?

As of the current stable release (version 0.14.23), this is not possible without the use of mods.

There appear to be a few mods which can do this - Launch Control and Auto Launch. However I have not used either of these.

In the latest experimental releases (version 0.15.xx), a new option "auto-launch with satellite" has been added to the rocket silo, as described in this developer blog.

The rocket silo now has an auto-launch checkbox so you can launch them automatically, and the launch is only going to happen when you insert satellite.

The latest version 0.15 builds are publicly available as "experimental" releases, which you can download from the website (if you bought the game there), or by opting into Factorio beta releases on Steam (further details on how to do that here).

At the time of writing there is no word from the developers on when version 0.15 will be moved into the stable release branch.

Yes, as of version 0.15 it is possible. See from the release page here

Release notes include:

•Added the ability to auto-launch the rocket.