From bagdogra travel to Siliguri by cab. It's a 30mins journey, won't cost you more than 100 bucks. Then you can hire a shared Tata Sumo or Bolero to Gangtok.
Yes, you can easily book NE Share at 699/person which goes daily from Bagdogra to Gangtok via Siliguri. To book or view details : www.ne.cab or call at 7872929029
Yes. Go for Wizzride. It's Simple, reliable and trusted by the Gangtok locals. Over 1500 Google Reviews and Rating of 4.7 Stars. You can download the Wizzride App and make a booking in advance.