LTA includes Helicopter Journey?

Leave travel allowance is exempt from income tax within certain limits as under : -
(a) Where journey is performed by rail; railway-fare in Second AC class by shortest route to destination.
(b) Where places of origin and destination are connected by rail but the journey is performed by any other mode then Second AC class fare by shortest route to the place of destination.
(c) Where place of origin of journey and destination, or part thereof, are not connected by rail and journey is performed by any other transport; then
(i) If a recognized public transport system exists between such places the first class or deluxe class f are of such transport by shortest route, or,
(ii) If in other case, Second AC class fare for the distance of the journey by the shortest route, as if the journey has been performed by rail.

Now answers to your questions:
1. If there is a public transport route exist between Bagdogra and Gangtok, either road or train. then only fare of that will be exempted.
2. If any public transport exists then tax rules does not allow taxi expenditure for claims and only fare on that existent public transport will be exempted at maximum.