to allow instantaneous or near-instantaneous travel
If I remember correctly, I think I read it somewhere probably "The Treatment" or "Talon Rogue" stories, traveling through a Warp Gate takes about 15 minutes.
do ships themselves actually have faster-than-light travel
No, ships can travel large fractions of the speed of light (the Z14 and Talon Rogue can go .3c for example. The fastest known Light Drive can go .9c). The Warp Gates are for FTL travel. Such relative velocity for such small ships means that it can't be from a conventional Newtonian dive, hence the Alcubierre Drive.
I thought the gates puttered out to their positions at sub-light speeds over the course of centuries
Correct, that's why Planet Rushes are 200 years apart, because that's how long it takes the gates to reach their destination. If we were to speculate, 200 years apart x14 Rushes = the Planet Rushes have been going on for close to 3 thousand years and the radius of explored space is also about 3 thousand light years (quite a small amount since the galactic radius is 50 thousand).
FTL travel wasn't really possible. In TiTS
True, except for the Warp Gates. Anno tried to miniaturize them for personal use a few years ago. Didn't work very well the first time(this is why Syri got her genitals swapped) but what else could you expect trying to take something from massive space infrastructure to personal sized in one go. Though this could mean personnel teleporters for end game content.