1 It is not necessary to have a company registered in India but you may do the business of training as a branch of your concern in Middle East or even you may have a franchise / associate in India.
2 In my view you may link your NRO account through the Indian payment gateways, as they long as they accept it. From tax point of view, the income will be treated as income earned in India and will be taxed accordingly. Further, as you are a Non Resident, the payees will be under obligation to deduct tax at source. It may be very tedious and inconvenient way of handling your Indian affairs, as the payees will have to comply with the TDS provisions, which many will find difficult to comply with. So from this perspective, you need to have an Indian establishment which will be a taxable entity under the Indian Income Tax Act.
3 You will have to register for professional tax with the state in which you will be carrying your business.
4 You will be liable to service tax, if the value of taxable services exceed Rs 10 L.
5 In view of 2 above, I suggest that you should have an establishment in India and invoice the students through such establishment.