Yes, the pulse dye laser is the preferred treatment for spider veins on the face.  The pulse dye laser targets the blood vessels directly without damaging the overlying skin and works quite well for facial spider veins.

Rosacea is best treated with the V-Beam or pulse dye laser.  This is a laser that passes through the overlying skin and targets especially the redness and broken blood vessels common in rosacea.  Laser resurfacing may actually cause a flare-up of rosacea given the inflammation to the overlying skin.  It depends on how severe your rosacea is.  If you are looking to treat that alone, I would use the pulse dye laser.  If you are looking to treat wrinkling and sun damage and are trying to do a laser resurfacing for those issues, I would have a discussion as to whether this is the best treatment for you given the rosacea.

Yes, it would be fine to resume your sessions now ensuring that you had pigmented or dark-colored hair. The laser does not work for gray or blonde hair. Hairs that are in the growth cycle are usually removed permanently, but hairs that are not currently in the active growth cycle when treated with the laser tend to become more fine with each treatment.  Likely this is what happened to you. Now you still have hair but it is finer than it was originally.  And yes, you would likely need fewer sessions to complete the hair removal process than someone just starting out.

Yes, fillers particularly Restylane, are very effective for treating hollowness under the eyes.  This is something we do very often and we have been very happy with the results.