I always imagined he was dead. I suppose he could be elsewise, but I think Lord Seeker Lambert is either dead or controlled somehow; if we do meet him, I doubt he will be himself.
The templars would only have authority to do so if they were acting under the Chantry's name.
This is untrue. The Templars were formed outside of the Chantry (this is the premise Lord Seeker Lambert uses to rebel/separate and it seems to fit with historical records we have) and chose to join forces with the Chantry to take control of the Mage situation, under the Chantry's purview. As the Mages have officially rebelled against the Chantry and thus the Templars are no longer needed within the Chantry, I believe the Templars can become an independent organization again. Whether this is allowed as legal or not likely has more to do with individual governments than anything else. I think it's foolhardy and Lambert's being a dick about it, but I do not believe we can prove they have no independent authority since they did exist as an external force at one time.
Justinia let them vote on independence, did she not? And even then, nobody attacked anyone until a dumb templar stabbed a mage who was surrendering peacefully. Then all hell broke loose.
No. Justinia let them have a conclave to discuss the situation with the Rite of Tranquility. The mages (some extremists) pushed the idea of discussing independence. Which they had previously been allowed to vote on but not since the college had been suspended and which they'd basically been warned against doing and knew would be seen as treason. Once the discussion began, the more conservative mages were even coerced into continuing and favoring independence (whether they would or wouldn't have at this point is unclear) by rebellious mages pointing out they would already be seen as treasonous just for attending the discussion.
Then, fighting broke out - which I do not condone (Lord Seeker Lambert, though not without skill and a point, is a brute) - and the mages defended themselves, very purposefully not hurting the Templars, but using magic. A (dumb, young) Templar likely didn't *realize* the mage was surrendering (this is how it seems from the point of view it's told) and attacked, out of fear. How often do you think Templars face so many skilled and trained mages? Just one unleashing all his/her power, even without the augmentation of blood magic, can destroy so much - this is not unreasonable fear. Then, that necessarily escalated.
It was Lord Lambert's doing that this situation occurred; I wouldn't blame individual Templars who were likely terrified, with reasonable fear. The extremist mages are not without blame in pushing things too far. Even Rhys agrees on this.
Remember too - while it initially seemed Lambert had faked Rhys's framing, it was made clear to us that he did NOT frame Rhys and had no way of knowing that he wasn't really the murderer. Rhys didn't exactly go to him early, he never produced Cole in time, and the knife that killed Pharamond was planted in his room by Adrian (or whatever her name was). Seeker Lambert is an overzealous dick, but it wasn't 100% his fault. Others are culpable as well. It was the Libertarian's (or at least Adrian's, but she didn't seem to be alone) ambitions that such a thing would occur - maybe not the massacre, but something to compel the more moderate and conservative mages to side with them on the issue of independence. As to Lambert, he seems like a man desperate to regain control the whole novel. He's a villain to be sure, but not an unsympathetic one when you consider what he has seen and what he truly wants (which is not all the mages dead - in fact, he clearly wants to find the murderer and protect the mages in some instances).