Three years later and I've had the same problem. I downloaded a vector sheet filled with multiple objects to be used. I only wanted one object off a sheet of ten. So I wanted to cut and paste that object into another svg file I was working on. Here's what I discovered and how I did it.
The original svg file with multiple objects on it was a single layer. You MUST create a new layer in order to do this otherwise nothing will work.
Create a new layer above the original layer, while in the new layer create a shape over the object you want to cut out. I used a rectangle with no fill and a solid stroke so I could see it around the object I wanted.
Select Path, Intersection.
Now go back to the original layer where your object resides, otherwise you will just cut and paste your rectangle instead.
Put your cursor over the object (In selector mode) and the object will now move independently of the original image.
You can now copy and paste this object into another file and it will retain it's attributes.
The biggest difficulty with Inkscape is getting the processes of GIMP or other image editors out of your head because this program needs a different approach. Instead of just using a selection tool then copy and pasting. You need to create a selection mask with a new layer and the shape of another object. Then the intersection tool will create a clipping mask for you.