it depends on ur choice to what to choose but it will be better to join indian navy as u r having ncc certificate u will be having reservation.
yes u will be eligible for BNS after doing bsc.
institute which offers courses for merchant navy are:
aqautech insitute of marine studies, new delhi
maharashtra academy of naval education and training, pune
vles institute, chennai
chennai school of ship management, chennai
there is separate forms available for these institutes which allows u to sit in written exam
after qualifying exam u will be examined for ur personality and mental ability then there would be medical examination or verification of cerficates produced by u.
fees for course is approx rs750000