1- Captain America is chased and almost killed without anyone really questioning it. The "World Security Council" is Ok with it. At least they could have incriminate him in a convincing way. A National hero is chased and nobody gets involved.
Again, Agent 13 and others questioned it. It took a direct order from a high-ranker to drown the protests. The council wasn't even in the picture because it's an internal affair. Cap is a national hero AND a SHIELD agent. HYDRA was pulling the strings behind the scenes as well under the guise of national security, closing down lines of communication.
Pierce used the truth/facts of the situation against Cap: that Cap was hiding something about Fury's death and was on the run. If anything, it's a great example of good writing.
2- Captain America, believed as being hiding something about Fury´s death, appears from nowhere, full of conspiracy theories and everyone instantly believes in him. This tells us this Council believes in everything and nothing ever gets questioned or investigated. The launching could have simply been delayed and things investigated a little better, but no. The answer is "You sick son of a b***. Arrest him.".
Have already explained why they believed him. Sure it's a conspiracy theory, but it's a secret upper echelon of power that deals in such political play daily so it isn't out of the realm of possibility. And the bearer is none other than Captain America.
If you rewatch the scene during Cap's speech, you can see the council members turning to Pierce when Cap said he's the HYDRA leader and see Pierce cocking his head in acknowledgement that Cap was speaking the truth. The launching was going to be delayed but HYDRA started moving and forcing it. See the launch room scene with Agent 13, Rumlow and the launch operator.
There was already an underlying tension between Pierce and the other council members if you remember. The other members had a private discussion about the tenability of his tenure. Seems like it wouldn't take much for the council members to call Pierce an SOB.
3- Security concerns should work both ways. If you don´t know what a man is up to, you shouldn´t give him the power to decide what happens and what doesn´t happen in an agency like SHIELD. It´s an intelligence agency. Maximum security should be a priority. You simply don´t make big moves based on trust. That´s not how a real intelligence agency works.
See above about Pierce acknowledging he's the HYDRA head.
4- The way Captain discovered the whole thing about HYDRA is lazy and dumb.
Well, I'd agree it's somewhat lazy in that it's delivered in the form of an exposition dump. It's probably the most out-of-place scene, but it's done so for pure comic-book, pulp sensibility's sake. It's hardly dumb because it references a very real-life event in Operation Paperclip.
For a movie who deals with intelligence, there´s really not that much intelligence involved.
Not a fair statement since a lot of your gripes have been explained.