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How was Agent Carlin still alive in the end of the movie when his (1 or 2 days) older version who came back in time to prevent the murder and the blast died while doing it?
Also, he wouldn't have known about this super-secret and serendipitous ability of the government to visit and send things in the past if he wouldn't have been investigating the blast and the murder. He prevented those things by using the time travel.
Basically, What about the Time-Travel paradox? Is there an explanation?
You seem to be requiring the movie to follow Back to the Future/Looper rules, where something that happens to your earlier self "ripples out" to your current self, or you think there has to be some sort of "stable time loop". The exact mechanism of the time travel isn't pinned down, but it's clear that each version of a person is a separate person. One interpretation is that the government has discovered a portal to a parallel universe that is four days behind ours. The Carlin we follow through most of the movie goes to the parallel universe and dies. The Carlin of that universe is still alive. Carlin finding out about portal, etc., are all things that have already happened in the "original" universe (although this doesn't seem to be the original original universe; this universe seems to be interfered with by another universe four days ahead of it, and that one probably was interfered with by another one, and so on), and nothing that happens in the parallel universe affects that. There are various other ways of describing it that are equivalent.
As is clear from the links shared by @BCdotWEB, there are paradoxically 2 Carlin's existing at the same time as a result of Time Travel- present day Carlin and future Carlin. The story involves the formation of multiple timelines, with the actual picturization of only the final timeline which is filled with the changes brought in by the previous time travels and multiple hints for the audience to figure out that there have been other time travel attempts before this one resulting in other timelines. The final timeline is successful in the sense that it prevents the crimes that happened in the other timelines. For detailed and a much clearer explanation, I'm reposting the links from BCdotWEB's comments above-
https://upload.wikimedia.org/wikipedia/commons/b/bc/Deja_Vu_Timeline.png
