When I run the program from the engine, it looks like this: http://adityakulkarni.com/astar/scrn.jpg
But the assets in the actual webplayer look completely skewed:
I am a bit new to unity, would appreciate some good suggestions.
Because your editor is in free aspect mode. To see what it will really look like you need to toggle it to WebPlayer.
On your first pic, top left below scene, it shows free aspect. Open up the drop down menu and select Web Player. It could be Standalone.
If you need to modify the size, go to Edit->ProjectSettings->Player and modify the size from there.