A story is a thread of causality from a problem to its solution. There is no story without some impetus, and that impetus is almost always going to be a coincidence on some level. This impetus creates some problem for the hero, and they act against it somehow, and the world reacts to their action, and so on and so forth until the problem is resolved in some fashion.
All but the simplest stories aren't just one story, but many substories, so chances are you're going to have coincidences in the thick of things which are really the impetus for some subplot.
As Rhodeworks said, you can use coincidence to get the hero into trouble (i.e., to start a story), but you cannot use it to get out (i.e., to resolve a story). It's a broken link on the action/reaction chain.
Readers have to accept the premise to be invested in the story, and oft times some coincidence is going to be a part of that premise.
This is the start of a story, and fine.
This is a hm coincidence, but if both these scenes happen near the very start of the story, I could buy it. Again, it's the premise of a story, which gives you some leeway.
and her path to the castle takes her straight past it (not a coincidence, but it looks that way at first)
This could look bit dubious. If you aren't already, I'd advise perhaps foreshadowing that it isn't a coincidence. Or give T'vaskli some reason to take this path.
(It's not clear whether "it" is the attack, the castle, or what.)
Again, the start of a story, and mostly fine. His learning of the crime lord's plan could be iffy depending on execution. The part that provokes a question from me is why T'vaskli? Are there not other pyromancers? How did he find her specifically?
The attack happens just like it was predicted, Falquin uses the Magnus to fight back just in time for Tenlash to see him and decide to take him on as an apprentice. Thus, the rest of the story happens.
Could be a coincidence, could not be. It seems clear to me that Tenlash could easily be in the town, since he already knows of the attack, and if there's a dude fighting in a giant robot, who isn't going to see that?