I doubt Solas would tell Leliana his real age, only the one that will mislead her more. Neither does your points actually gear Leliana with enough clues to accurately discover Solas' lies. The fact that he became a hermit at some point away from civilization simply boggles hundreds of possibilities of who he could be, where he's from, which Leliana is simply not omniscience enough to find out. And according to Gaider, the roughly estimated the time period of the Inquisition is 3~4 years, which makes sense story wise considering the travel distances between each areas. There's a forum about it somewhere here.
And I think at this point you're simply expecting some kind of impossible perfection from her. Spies can get into Inquisition and it's not something Leliana can completely put to an end. She doesn't know, and can't, control people's loyalty and changes and the best she can do is to discover them at quick notice once a suspicion comes up. And I don't know exactly what Solas did to instigate this kind of immediate attention from her. Plus you've seem to have forgotten that it was Leliana's agent who saved Josephine life, as a counter to the potential Repose assassination attempt.
I'm not entirely sure why you're complaining about her incompetency and then turn the other cheek to say it's not competent for a spymaster to do shady business like assassination, torture, crippling people from shadows, shady dealings etc... Which is exactly what she dealt with as a bard, and exactly what Divine Justinia used her for. Also Her dark ending is subjective, some prefers the Divine to be diplomatic and peaceful while others prefer her to be vicious and iron-fist-ed (like Vivienne) so it's the matter of personal opinion, not the character's competency. So yes, Leliana isn't perfect but neither is she incompetent.
As for the Redcliff future you're being a bit dramatic about it. The world was already doomed, Alexius (who has already gone too far) was going to receive death sentence from Coryfish and somehow killing his long gone blighted son puts the entire world into jeopardy? (I mean this can be counted as a mercy killing too). I personally tried to stop her in my canon play through but if you've paid enough attention to game, a year of tortures and experimentation have broken her beyond repair. Surely you can tell by how she states that 'no one is innocent' before silting Felix's throat (that broke my tiny heart) :c It's not even related to her being a spymaster so I'm not sure what your point is.
Really, some people expects her to be like an all knowing god for some reason :/
That's my point. If Solas declines to reveal his age or misleads her more, then it confirms that he can't be trusted. But Leliana did not ask the right questions or investigate this nearly enough. Gaider was referring how long it took to make the game. He misunderstood the question he was asked, and he has since clarified this. He did not say Inquisition's events happened over the course of years.
Yeah but I don't see Leliana stopping any of them. The Qunari can discover the spies in the Inquisition much better than she can, and the Qunari are only working through agents. And yes, it was Leliana's agent who protected Josephine, not Leliana herself. But so what? Josephine was being targeted for assassination. Posting bodyguards is only natural. What's her alternative, not to post them?
She's competent as a bard, but a good bard doesn't make for a good spymaster. The endings are proof that she needs the Inquisitor to define her. Vivienne? No. Cassandra? No. Only Leliana's method of leadership is subject to change based on the Inquisitor, and if the Inquisitor does not get involved with her for her personal quest, then she is indecisive and leads the Chantry to the brink of destruction. That proves she is incompetent.
Yes, killing Felix at that point put the actual world in jeopardy, because that set Alexius off and made him attack. If his attack had been successful then the Herald fails and the future takes place. This is compared to Leliana listening to the Herald, letting Felix go, and then having a strong possibility of Alexius simply turning the amulet over to Dorian without a fight. The Herald had the upper hand. He could have easily convinced Alexius that him and Dorian going back in time to prevent the future from happening would make up for his mistakes. Alexius, having been marked for death by his master anyway, would have agreed. But no, Leliana had to let her emotions get the better of her, and for no good reason. And because of that a tough battle ensued, a battle that could have cost the world everything.
Mercy killing? Felix isn't even supposed to be alive at that point. None of that future is supposed to happen. That's the point. Our goal was to get the amulet back and go back in time, thus preventing Felix from getting that way. We weren't there to mercy kill, or anger Alexius, or make empty proclamations about how we want the world back or how no one is innocent. If Leliana wanted to make herself useful, she would have stealthed up to Alexius and stole the amulet from him without him noticing. But no, she takes a hostage, and it wasn't even Alexius that she took from behind, it was Felix. And then what did she do once she had him? Killed him, thus destroying the only leverage they had over Alexius.