Exam

Yes, you will be qualified to register for JoSAA counselling with a 60% score in Class 12 for the colleges that don't have a 75% cut-off criterion. Though a minimum of 75% in Class 12 marks for general category candidates is stipulated by IITs and NITs and by 65% for candidates belonging to SC/ST/OBC and physically challenged categories, there would be eligibility criteria different from that for each of the colleges. If the institutions you are targeting do not adhere to this cutoff, you may participate in JoSAA counselling provided you have qualified in JEE Main or JEE Advanced and satisfy specific criteria set by those institutions. 

Eligibility would depend on merit requirements that demand a minimum of 50% in Physics, Chemistry, and Mathematics (PCM) scores from the respective 12th-grade result. Choose the colleges in the same regions of Delhi/NCR by finding available colleges that conduct direct admissions with some under the categories of Management quota or Vacant Seats. Contact these colleges directly or on their websites for their particular procedures of admission, which is usually submitting an application form and other supporting documents. Under the management quota, the fee will be higher. Following your application, keep in contact with the admissions office regarding updates on the status of your application. 

You will surely benefit from your home state quota and OBC NCL for admission to DTU via JEE Mains. Home state quota is the admission of those students whose 12th grade they have done in that respective state. Preference is provided to the candidates during their admissions, and they get lower ranks compared to others in the case of home-state quota admissions. In the case of an OBC-NCL, he will benefit from the reservation through quotas, which is, as defined, a percentage of seats reserved for the exclusive advantage of the OBC candidates. If you are eligible under both quotas, you benefit at the same time through them during the admission procedure. 

You can sit for the IISER exam or the IISER Aptitude Test (IAT) even if you passed your 12th grade seven years ago, as there is no age limit mentioned for IISER admissions. But the eligibility for the exam includes that you have to be an equal pass out with at least 60% marks (for the SC/ST/PwD candidates, the cut-off percentage is 55%). Your main qualification has to be in Biology, Chemistry, Mathematics, or Physics. At least three of these Science subjects should be studied by you in the 12th standard. Hence, provided you have met these academic requirements, it does not matter when you completed your schooling; you are eligible to apply for the IISER exam.