Screen shots would be very helpful here...
In general, though, you should use File > Place.. to import your illustration rather than using copy/paste.
Part of your problem may be a display settings issue. Make sure you have your view settings (View > Display Performance) set on High Quality Display.
You should get better results using File > Place...
I would move that to its own artboard in Illustrator first, though.
Curved lines are likely to render jagged in the InDesign Preview, especially with Typical Display instead of High Quality. This does not affect output.
In Illustrator, I would view the object in Outline mode. There, see if those extra invisible boxes exist, and if they do, delete them the direct select tool. Once your artwork is all cleaned up, copy and paste it into InDesign.
I wrote an article awhile back about copying and pasting from AI to ID, and dealing with those invisible boxes: http://indesignsecrets.com/follow-the-yellow-brick-road.php