Could Cooper from Interstellar see himself as he was in the Tesseract had he searched a little more? If he can see the past, he could have seen himself in the present, and probably in the future too right?

He could... but that would have taken the movie in a different direction...

He was sent there to communicate with Murph and this whole set up was done by "they".

******** Who were "They"? ***********
 The human species gets suffocated due to blight and become extinct. The only survivors were some scientists and engineers, probably at NASA who survived cause they stayed underground.
 They carried on with their research underground and lived that way and carried human species forward for years to come, and may be after a few decades or after 100 years or so, they were able to access the higher dimensions, by using a tesseract and placing it near the singularity of some super massive blackhole, cause at singularity the time stops and hence, they could have access to infinite space and time, and that allowed them to access the higher dimensions, the 4th and the 5th dimensions. So, these survivors were "They", the so called beings of the 5th dimension.
 "They" found a new home for human beings of the past, the Edmund's planet in a distant galaxy, and placed a worm hole near Saturn to help them reach there.
 Now, they needed a person from the past, smart enough to whom they could give the quantum data using gravity, who could solve the equation of gravity.
 They must have shortlisted a few such people (Murph being one of them), but there was a PROBLEM..
 ******* THE PROBLEM *********
 Sending the quantum data using gravitational anomaly to these chosen people would have worked only if
 1. that person was a physicist working on the equation of gravity. ****************** CONDITION 1
 2. the data would come from someone with whom this person shared a strong bond of Love. ******************* CONDITION 2
 Only once these 2 conditions are met, that data would make any sense to that chosen person.
 1. they needed to get this person to NASA so that he/she would be aligned with task of resolving the equation of gravity.************ to tackle CONDITION 1
 2. they needed to get someone, with whom this person shared a strong bond of love, inside the tesseract near a singularity, so that this person would use gravity to send the quantum data from inside the tesseract, to the chosen one on earth hoping that the bond of love they share, will do the trick. ************ to tackle CONDITION 2
 So, out of those chosen ones, they chose Murph cause she shared that strong bond of love with her father who also happened to be an Astronaut with NASA, and he could be somehow lead to the tesseract in the outer space, and they felt that they could use this father-daughter duo of Joseph and Murphy Cooper to make this mission a success.
 We were shown that after the crash Cooper left NASA and became a farmer and later he got the news that NASA has been shut down.
 Later when NASA was Re-instated as a Secret organization, cooper was not aware of NASA being still operational.
 He receives NASA coordinates through a gravitational anomaly and He and Murph reach NASA n there he finds out that NASA is actually still operational.
 Cooper goes on the Lazarus Mission through the wormhole and jumps into the event horizon of Gargantua.
 ******* THE PARALLEL UNIVERSE *******
 Remember Murphy's law, everything that can happen will happen. So, when we were shown that after the crash Cooper left NASA and became a farmer, in some parallel universe he continued to be an Astronaut with NASA even after NASA became a secret organization. And cause they had access to infinite space and time and the 5th dimension, "they" could observe that parallel universe too.
 So, it turns out that "they" could observe both the versions of Cooper in 2 parallel universes.
 1. Cooper(the farmer)
 2. Cooper(the Astronaut)
 In that universe too, they placed a wormhole near Saturn and Cooper(the Astronaut) captained endurance through that wormhole, and as per Murphy's law, in that universe too if he would have a choice of jumping inside Gargantua's event horizon, he would.
 Christopher Nolan gave mission the name "Lazarus Mission", which implies someone had to die and then come back from the dead.
 The big secret here is that the Cooper (the Farmer) that we see jumping inside the tesseract in the movie dies as soon as he jumps inside the event horizon.
 And the Cooper that we see later inside the tesseract is the Cooper(the Astronaut) from the parallel universe, and he is the one who is dragged into the parallel universe of Cooper (farmer) inside the tesseract by "They" and he is the one who later reaches the Cooper Station.
 So, Cooper (the farmer) dies and comes back from the dead, from a parallel universe as Cooper (the Astronaut). that is why Christopher Nolan chose the name "Lazarus Mission" for this mission.
 ******* THE SWITCH *********
 The switch from Cooper (the farmer) to Cooper (the Astronaut) was made when Cooper (the farmer) jumps into the Event Horizon of Gargantua.
 There was an untimely and relatively unnecessary Cut sequence there, where the scene switches back to earth, for apparently no reason.
 Here's how it went. In that scene we see:
 1. Cooper(the farmer)'s space craft catching fire soon after he detaches from endurance. (Location: inside Gargantua)
 2. Cooper(the farmer) in pain (Location: inside Gargantua)
 ------ CUT SEQUENCE -------
 3. Cooper(the farmer)'s corn field on fire (Location: Earth)
 4. Cooper(the farmer)'s son watching helplessly in despair. (Location: Earth)
 5. Murph filled with hopelessness, looking at the broken replica of the moon lander that her father gave to her. (Location: Earth)
 So, here everything that belongs to or is related to Cooper(the farmer) is shown as kinda broken/on fire, indicating that Cooper (the farmer)'s world is coming to an end and this is where we say good bye to Cooper(the farmer).
 ******* INSIDE THE TESSERACT *********
 Right after that cut sequence the movie makes a switch from Cooper(the farmer) to Cooper(the Astronaut) who jumped inside Gargantua may be decades ago.
 So, when Cooper(the Astronaut) jumped inside Gargantua in that parallel universe, at that time, Cooper(the farmer) was probably living on that farm with Murph.
 The Cooper that we see inside the Tesseract is the Cooper(the Astronaut) from the parallel universe who has been dragged into the universe of Cooper(the farmer) inside the tesseract by "they".
 Here, upon seeing Murph from inside the tesseract he realizes that he is looking at a Murph from a parallel universe and initially he feels that he has failed in his mission, so he feels that in that universe of Cooper(the farmer), he should make them STAY together, hence he sends the STAY message.
 Then after TARS (not the TARS we saw earlier, this is TARS from the parallel universe of Cooper(the Astronaut)) tells him about the tesseract, Cooper(the Astronaut) eventually realizes that "they didn't choose him, they chose Murph to save the world".
 Now everything becomes clear to him, and he,
 1. gives out the NASA's coordinates to Murph and Cooper(the Farmer), so that Cooper would take Murph to NASA ********* TO FULFIL THE CONDITION 1 **********
 2. gives the quantum data through the watch. ********* TO FULFIL THE CONDITION 2 **********
 Once, both the conditions are met, "they" know that his job there was done and hence, they close the tesseract and send Cooper(the Astronaut) back through the wormhole after which he is rescued and taken back to the Cooper station.
 Now as "they" have access to infinite space and time, "they" make Cooper(the Astronaut) travel through the wormhole at the time when endurance, with Cooper(the farmer) inside was going through the wormhole in the opposite direction. There Cooper(the Astronaut) distorts spacetime to create an impression of a handshake with Brand.
 So, when Cooper(the Astronaut) jumped into Gargantua, "they" dragged him into Cooper(the farmer)'s parallel universe,
 Why did they do it that way?
 Why did they not let him be in his own universe, of Cooper(the Astronaut)?
 The answer is cause that would never have worked out. If from inside the tesseract, Cooper(The Astronaut) would give out the NASA's coordinates back himself (Cooper(The Astronaut)) of the past.
 Then Cooper(The Astronaut) of the past would have known where those coordinates lead to, cause he worked with NASA (the secret organization), and that being a top government secret, he would have never shared that information with Murph and hence, CONDITION 1 would not have met.
 So, they had to create a parallel universe for Cooper in which he would think that NASA has been shut down and would not be aware of the fact that NASA was later re-instated.
 To make that happen, they create a Gravitational anomaly to crash Cooper's spacecraft (the one he had a nightmare about in the beginning of the movie) to somehow force him to quit NASA and hence, created a parallel universe of Cooper(the farmer).
 When Cooper(the farmer) received the NASA's coordinates, he had no idea what those coordinates lead to and accidentally ended up taking Murph to NASA, hence fulfilling the CONDITION 1.
 That for me is the Plot of Interstellar. There was no time loop or bootstrap paradox. In the 5th dimension there never is.
 And if you want to find the answers to questions like,
 - How did Murph figure out that the Cooper was her ghost?
 - What did TARS mean by the statement - "They did not bring us here to change the past"?
 -The clues in the movie that support the theory of 2 Coopers from 2 parallel universes.
 Watch this detailed video with the explaination of the whole plot along with all these answers by clicking the link below.
 Interstellar explained (along with the bootstrap paradox).

Not how the Tesseract manifested itself to him. Cooper was presented a seemingly endless array of moments in time – dioramas – relative to a common place, i.e. Murph's bedroom. There were in fact instances when Cooper could see himself in the past, and those were those instances when he was interacting with Murph, especially when he was on the verge of leaving for the voyage. But beyond that, Cooper's experience in the Tesseract was controlled, or orchestrated, by the FFDH's to allow that wonderful solution to come about. Cooper realized that the wristwatch was the one tangible object that not only be affected by gravity, but would be something that Murph would retrieve and keep with her, based on her relationship with Cooper (her father).

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