1)How does time travel in Interstellar work? ... Thus the event of him visiting NASA precedes all the Morse code communications from within the black hole.
This scene hinges on the idea that we don't know what happens inside a black hole (by "inside" I mean beyond the event horizon, which isn't really "inside" anything - it's just beyond the point that any information can escape to the outside universe.) Someone - some unknown agents from the distant future - sent this time machine back to rescue Cooper and allow him to communicate with his daughter in the past, because "they" figured out that this sequence of events needs to happen for humanity to escape the dying Earth.
COOPER (frustrated): Who’s ’They’? And why would they help us?
TARS (over radio): I don’t know, but they constructed this three-dimensional space inside
their five-dimensional reality to allow you to understand it ...
COOPER: It isn’t working -!
TARS (over radio): Yes, it is. You’ve seen that time is represented here as a physical
dimension - you even worked out that you can exert a force across spacetime -
COOPER (realizing): Gravity. To send a message ...
How this time machine works is not explained beyond the above, we just have to accept that it does and it allows him to send messages into the past from within the black hole.
As for Cooper's visit to NASA preceding the communications - it does, from Cooper's perspective. From Murphy's perspective, the communications come first, but she wasn't inside the black hole.
So how would he have gone to NASA the very first time? Even before his ghost is yet to be formed?
I know this is a little mind-bendy, as time travel tends to be, but you have to understand that Cooper's actions inside the black hole happen after he went to NASA, for him. They happen before he went to NASA for Murphy. The two time periods are being connected, like a piece of paper folded back on itself.
2) How is Cooper, from the black hole able to simultaneously send information to the teenage Murphy and adult Murphy at the same time? And what is the collapsing of the black hole before cooper passes out signify?
He wasn't sending information to present day Murphy. Present day Murphy was just remembering the messages from the past and finally connecting the dots. The scenes are interspersed to make the reveal that much more dramatic.
Scene 1: Murph's past
INT. MURPH’S BEDROOM - MORNING
Murph (ten) stands there, startled, STARING at the
bookshelves. At the book on the floor, a broken toy beside
Scene 2: Murph's present
INT. MURPH’S BEDROOM - TWILIGHT
Murph (forty) looks at the bookshelves, REMEMBERING ...
Scene 3: Cooper inside the tesseract (the time machine)
INT. THE TESSERACT - CONTINUOUS
Cooper watches Murph (ten) cautiously approach - she
CROUCHES. Picks something up -
3) While reaching Miller's planet, Cooper proposes a method to land on Miller's planet yet escape the time shift. But Cooper mentions to Brand and Doyle that 1hr = 7yrs. So why does the time shift occur anyway?
Cooper's plan wasn't to escape the effects of the time dilation entirely, just to find a place to park the Endeavour where it wouldn't be affected - so that those who remained behind could continue the mission, without losing time to the dilation effect, if the Ranger shuttle never came back.
However, the Ranger and those aboard would always have been affected by the dilation. It's not possible to enter that planet's vicinity without being affected.
4)It's evident that out of the three prospects, some of them are either too close to a black hole or some do not even sustain life. What mission does Mann keep saying that he has to complete
There are a few different "missions" Dr Mann refers to throughout his scenes in the movie. One is the mission to save humanity (which he intends to continue doing primarily by escaping this icy world by any means necessary,) another is the "mission" to survive, himself. He goes on a lot about instinct and the will to survive as individuals vs as a species.
There's also some discussion in the middle about how Plan A (to find a new home and evacuate Earth) was a decoy all along and the real mission was always Plan B (to find a suitable colony world and leave Earth behind.)