You can map the hospitals through on Google Maps using QGIS software. You can also work through other softwares like ENVI or ArcMap, but QGIS is best in this
Yes you cant do it at country level. These are micro level studies, you should choose either state or at district level. Even at state levels it is not easy for larger states like Uttar Pradesh or Bihar or other, but for states like Delhi, Goa or Kerela you can do it. If you are doing PhD on this you can work on it and take help of some juniors in digitizing it manually