Welcome to the forum!
Yes, I would suggest there are better vector tools than the eraser, to remove unwanted things. Inkscape's Eraser tool seems to have some bugs; I can't always get it to work. However, just to mention it, the red indications which the Eraser tool makes are not permanent and are not part of the image. They should disappear shortly after the mouse button is released. It could be a(nother) bug, if the red marks are remaining more than.....say 5 to 10 seconds after the mouse is released.
For the case you've described, where the ellipse and the rectangle should have the same width and are aligned vertically, I would use Path menu > Union. It will make the 2 objects one, removing the inner segments, automatically.
However, your drawing shows the ellipse is slightly wider and/or misaligned. If you were to use Union on them as is, the misaligned parts would become part of the new object, and probably that would not be acceptable. (Try it to see what I mean
) If there were some reason why you needed to join those 2 object, as they are, you would remove the unwanted segments, and then join the new end nodes together. Note that if they are still shapes (rectangle and ellipse) they will need to be converted to paths first, using Path menu > Object to Path.)
To remove the segments, use the Node tool
. For the rectangle, click once on the segment, to select both nodes (on either side of where you click). Then click "Delete segment between 2 nodes" button on the Node tool control bar. Looks like:
You could do that on the ellipse as well, except you would still have a node leftover, which still needs to be deleted. Click once on that node, to select it, then delete (either Delete key or Edit menu > Delete or right-click > Delete).
There is arguably a little faster way for the ellipse. Duplicate the rectangle. Hold Shift key while you click on the ellipse (shift allows a multiple selection). Then Path menu > Difference.
Whoops, I see Lazur posted while I was typing. I'll post anyway, since mine includes some different info. And yes, Lazur gives yet another way to "edit" the ellipse. Using Ellipse tool, you can draw the arc originally as an arc.
http://tavmjong.free.fr/INKSCAPE/MANUAL/html/Shapes-Arcs.html(One of these days I'll finally make a tutorial on Node Editing..... )
Edit - forgot joining the new endnodes. Still with the Node tool, drag a tiny selection box around the 2 nodes that you want to join (selects both easily). The "Join selected nodes" button on the control bar.