I want to make an elipse and a rectangle the same width, and then join them so that the union between them is seamless, with the middle of the elipse right over the edge of the rectangle.

Is there an easy way to do this?

First, enable geometric bounding box, so you can be sure the objects are the same width.  Edit menu > Preferences > Tools > Geometric Bounding Box.

Use the Selection tool to make them the same width.

Enable snapping, so that snapping center of object and snapping to midpoint of lines are enabled.  Then temporarily fill the ellipse, and you can snap it's center to the midpoint of one of the sides of the rectangle.  Info on snapping:  http://tavmjong.free.fr/INKSCAPE/MANUAL/html/Snapping.html

Or if they are truly the same width, you could snap the left or right handle of the ellipse to the left of right corner of the rectangle.

I'm guessing that next you want to select both, and do Path menu > Union.  I think that's what you probably mean by "seamless"?

You could also just use snapping for that (i.e. no manual fiddling with widths and without having to think about geometric vs visual bounding box).

By default Inkscape snaps to corners, so you can create an ellipse that perfectly fits the width/height of a rectangle simply by snapping to two diagonally opposite corners of the rectangle when drawing the ellipse.

Alternatively you can enable "snap to path", then you can snap to the sides of the rectangle while controlling the other dimension freely.