Will I be eligible for GATE exam after five years or not?

The answers to your questions are as follows.
1. Chitkara University is a private university. It is having a good name and almost 90% of the students are getting campus placement in good companies with good salaries and benefits. The integrated BE + MBA dual degree is also good and you can opt for it. This is a good course. Many MNCs are taking students from this stream.
2. If you want to do PhD from NIT/IIT, why should you opt for BE + MBA dual degree? You can do BE and then ME and then PhD. In this process, MBA will not give you any additional scope or chances. BE + MBA is good for students who want to start their careers in a good industry. So if you are interested in PhD, take admission in BE only and concentrate on the subject. You can save one year. After that, you can opt for your PhD.
3 Now you can't attend for GATE examination. You can attend the GATE examination when you are in the 3rd year of your undergraduate course. As now only you are taking admission you have to wait for another 2 years for applying for GATE.
4. GATE scores are valid for 3 years. You can apply for admission to post-graduation courses or for any jobs where the selection is based on the GATE score for three years from its issuance. After three years it is not valid. If you feel that you got a less score, you can write the exam again for the second time and you can use any of the two scores you obtained.
5. Join in the course now and write the GATE examination when you are 3rd year of the course and if necessary when you are in the final year also.

Relating to your queries, the following responses could be seen to resolve your issues- 1) Chitkara University is a privately managed University but has its own reputation in terms of excellent placement of the students. The students undertaking the different courses are recruited through the campus placements long before their passing out the integrated BE +MBA course because of the quality teaching of the faculties apart from providing them sustained motivations. As a result, the top multinational companies would like to offer decent packages to the bright students undergoing studies from this university.
2) If you are interested to carry on your doctorate degree from NIT/ IIT, you should have opted for B. Tech and M.Tech for these institutions and later you can enrol your self for the doctorate programmes to be taken from one of such institutions.
3) GATE scores remain valid up to three years but since you have opted your dual course, writing for GATE test after completing this dual course would hamper your timing for the said test, though there is no constraint for the same.
4) Ensure that your percentile scores are impressive to be inducted for your doctorate programme and to achieve success in your attempt, go in through the guidance of some professional institutions providing you guidance at different stages.

Chitkara University is having many colleges under its fold providing many courses in different disciplines. It has a very good campus recruitment process and many meritorious students have been immensely benefited by that. They have global associations for many courses abroad. They offer a large number of UG, PG, and PhD programmes.
Going through the credentials of the Chitkara University we find that it is a good institution and your course selection for integrated BE+MBA is alright. Doing an engineering degree plus management qualification makes a person eligible for a job in the industry. If you have that idea in your mind then it is definitely a calculated decision and there is no problem in going ahead with this proposition.
Now coming to your ambition for doing PhD from IITs you have to have a good score in GATE which of course you can attempt only after your graduation and afterwards completing your post graduation in a subject where you have an ambition for research. If your present course selection that is BE+MBA is clashing with that objective then it means that there is a need to re plan your educational path for achieving the overall objective of doing PhD from reputed institutions in India preferably from IITs.
In that case you might have to do the UG course separately and then after seeing the result of that you would finalise the subject that you might opt in your PG.
Do you feel that after doing your PhD from IIT you would get a good break and make a good career? Please note that only doing something from IIT does not guarantee that one would have a great career. There are many other factors which help a person in securing a good career. So you have to see the pros and cons of all the options available and take a decision in the matter accordingly.
My humble opinion in the matter is that one has to do hard work in the course one has selected and all the good institutes in our country are having very good campus recruitment facilities and what matters is not only the institution but the merit and score that a student is able to achieve. For getting admission to the highly coveted institute sometimes there are some entrance tests or requirement of a high grade in the GATE or some equivalent exam and if the student is not able to perform well there then his admission to a top institution becomes a difficult proposition.
We should not mix too many things in our academic pursuit and whatever line we select we have to go with full force and vigour in that only. Having too many options and jumping from one to another would only result in loss of precious time. If one is able to get an educational path in one's interest and liking area then making it the ultimate aim would be a better strategy.