I already googled a face-on galaxy and an edge-on galaxy, but I couldn't find any proper information. Would you explain them to me?
The terms refer to the viewing angle, i.e. from which direction do we observe a (disk) galaxy. If we happen to be located roughly in its plane of rotation, we see it from the "edge", whereas if we are more or less above or below its plane, we see its "face".
To make a more quantitative statement, the orientation is described by the inclination angle $i$, which is the angle away from "face-on". That is, the inclination of a galaxy that is observed face-on is $i=0$, and the inclination of a galaxy observed edge-on is $i = 90º$. For randomly oriented galaxies, the mean inclination angle will be around $i = 60º$.
Wrt. your second question, you're right that the text is not correct. The three small constellations lie north of hydra. "North" means "toward higher values of declination". Perhaps it's more easily seen in this image (from Wikipedia):
I just want to add pictures to augment pela's answer.
Just google "edge on galaxy" and "face on galaxy" to find results like these, it is entirely about the viewing angle.