Firstly, does a mechanical shutter in a digital camera work the same way as in a film-based camera?

Yes, it's exactly the same.

If I use a slow shutter speed, am I actually exposing the sensor for a prolonged period of time or is this more like capturing a video and then layering the frames on top of each other?

The former. Shutter speed determines the duration of the exposure of the sensor. Many digital cameras can record video, too, of course, and in that mode they capture a sequence of distinct frames, but I don't think that's what you're asking about.

Secondly, do mechanical shutters provide a global shutter for CMOS sensors or does one still get a rolling shutter effect? Again, what about very slow or very fast shutter speeds?

The mechanical shutters you find in most DSLRs will give a rolling shutter effect at high speed, because the shutter only exposes a small part of the sensor at once as the first and second curtains travel across it with only a small slit between them. At slower speeds there's a period where the entire sensor is exposed at the same time, after the first curtain finishes moving and before the second curtain starts, so the effect is more like a global shutter. The shutter operation doesn't change -- only the time between the movement of the two curtains does.

There are also other types of shutters, like leaf shutters. These are pretty uncommon, but not unheard of. And of course there are electronic shutters, which can also be divided into rolling and global categories, depending on how they're designed.