Is it possible to draw this picture without lifting the pen?

It is possible, and it's actually quite easy. Start on the red dot and move your pen according to numbers in a picture below.

Here is one way, out of many.

As you can see, I started at one vertex and ended at the same vertex. Since the graph is connected, and every vertex has an even number of segments joined to it, you can start at any vertex and end there, covering all the segments.

Yes it is possible. With problems like these, look at all intersection points of segments. If there are $0$ or $2$ intersections where an odd number of segments join (including endpoints which are considered $1$ segment), then the task is doable. If more than $2$, it is not. Your drawing is definitely doable since all intersections involve $2$ or $4$ segments joining.

Similar problems have plagued my existence since third grade. One day I stumbled across a book on it when I was eleven or twelve and it provided a similar explanation to what I typed above.