Vascular surgeons are concerned with arteries and veins that are not in the chest (this is the realm of the CT surgeon). So, vascular surgeons do things like reaming out occluded carotid arteries (CEA's), bypassing clogged femoral arteries (fem-pop bypasses), doing AAA (abdominal aortic anuerysm) repairs (both open and endovascular), placing hemodialysis access loops, and many other things. This specialty is basically plumbing--if blood isn't flowing through an artery, you go in and fix it either by roto-rooting or bypassing the blockage.
Yes, they do complete cases from beginning to end. This is probably true for the majority of their cases. However, they are also very frequently called in on trauma cases that involved severe arterial compromise.
Vascular surgeons are arguably one of the most technically skilled surgical sub-specialties, although I'm sure many people will vociferously disagree with this. They have to be able to sew with incredibly fine suture material on small arteries, and their sewing must be as water tight as possible. If you do this as a career, get used to wearing loupes (these are the funny looking glasses that magnify). Also get used to being woken up in the middle of the night with lots of emergencies (acute arterial occlusions or ruptured AAA's need to be repaired immediately).
the vascular surgeons I worked with during my surg rotation loved it, but it is a long haul in terms of training--a 5 year general surgery residency, following by a 2(?) year vascular surg fellowship.
-mrp