your problem is in Unity. Go to the import options (click the original .blend you dragged and dropped into your project) on the rig tab, select legacy, hit apply, then on the animation tab, select what animation you want to be the default when the game starts, you can directly alter this in your C# or (icky useless) unityscript later
if the model has no animation saved. then thats the other problem. go into blender. open a dopesheet/action editor window. name the animation what ever you want in the action editor Idle01 or what ever. position the model like you want. select all bones, I to create keyframe. on the pose window in the far right end of the bar on the bottom you will see 3 page looking icons, these are copy, paste, and paste mirror. what you want to do is with the bones selected, hit copy (i think ctrl + c and ctrl + v work as well). then arrow keys or mouse to what ever frame number you want the animation to end on. and paste the pose. hit I again to make keyframes. then move back somewhere in the middle of that range of frames. and change the pose up a little. select all, hit I for keyframes. then hit alt+A to run the animation. that will serve as a quick and dirty idle animation. now save the file, delete the old one from your unity project, drag the new one in. set the rig to legacy. and see if the animation works properly now.
edit: since ive been down this road myself. i can tell you right now theres another issue you will run into in the future, that is one of axis rotation confusion. blender uses an unusual Z axis up configuration. while unity uses the conventional Z axis forward Y up configuration. this is not a problem in Unity until you have to use a Quaternion.rotation/lerp/slerp/identity on your model which will point the z axis in the unity z axis direction, usually resulting in your model laying on the ground face first. there is an addon called unity tools for blender that will fix the rotation for you. doesn’t apply to your current problem, but trust me, you will be happy that i mentioned it when one of your models takes a faceplant and you cant figure out why!