The application is compiled with maven. JUnit is used for testing porpouse. 

To compile and run tests:
$: mvn install

I assumed that the distances between towns are Integer. 

The class "com.trains.RailRoad" is the one which expose the methods to solve the problems described. 
This class contains a graph, which can be loaded with "com.trains.GraphLoader" class. To do this you must provide 
a graphString such as "A/B/5/D/5/E/7,B/C/4,C/D/8/E/2,D/C/8/E/6,E/B/3".

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I implemented a telnet server so you can use the application remotely. 
To run the server:
$: mvn exec:java -Dexec.mainClass="com.trains.remote.TelnetServer"  

Then connect to it:
$: telnet localhost 8088

You can execute differents command to the server:

help: It shows a helper which tells you the commands you can execute.
>>help

create: Creates a new RailRoad. Args: railRoadString
>>create A/B/5/D/5/E/7,B/C/4,C/D/8/E/2,D/C/8/E/6,E/B/3

distance: Calculate a distance for a given path. Args: routePath  
>>distance A-B-C

shortest: Calculate the shortest route between two towns. Args: town1 town2 
>>shortest A C

tripslimited: Calculate the number of trips between two towns with a maximum of stops. Args: town1 town2 stopsLimit 
>>tripslimited A C 3

tripsexact: Calculate the number of trips between two towns with exactly a number of stops. Args: town1 town2 stopsAmount 
>>tripsexact A C 4

tripsdistance: Calculate the number of trips between two towns with a distance less than a maximum. Args: town1 town2 maxDistance 
>>tripsdistance A C 30