That's definitely nothing to do with MipMaps, but something wrond happens with texture sharing - you are right. I don't know yet if this is in your code or somewhere in Xith3D, but we definitely should check this.
With your test, the sphere looses texture ONLY in the case when the bigger shape (opaque one) is visible, i.e. not culled.
Try to do the following: open the app, click on the big square shape, and using a,s,d,w keys move it off the visible area - you will see that texture on the sphere will appear.
Looks like this shape is rendered before the sphere, and causes wrong behavior of TexCoordGeneration [somehow?]. Note that transparent shape always rendering after the sphere and does not cause such problem.
So, as a bottom line, if it would be a bug report in IssueZilla, I would mark this issue as confirmed
What can I say - we definitely need to put this test in a simple test case, say, based on the test code from com.xith3d.test...