Yes, a Non-resident Indian can trade in equity futures and options.

Yes, NRIs can trade in the F&O segment of the exchange out of the rupee funds held in India on a non-repatriate basis. NRIs are required to get a Custodial Participant (CP) code to trade in F&O. 
Zerodha, have tie-ups for custodial business (currently with ORBIS), and a CP code will be assigned to you. Once you have the CP code, all the funds for trading F&O will sit with the custodian, similar to how all your funds for trading stocks sit with your bank.