I am trying to study the texture deformations caused when mapping a rectangle to a sphere, if I had a software that took an image as an input and displayed how it looks like when texture mapped to a sphere, it would allow me to observe and study this deformations better.
Although this is borderline "off-topic" for the site, there is perhaps a simple answer:
Why not use a webgl approach such as this one? http://learningwebgl.com/lessons/lesson11/index.html (parent page here: http://learningwebgl.com/blog/?p=1253)
Just rotate the image of the moon around until you have one of the poles visible and you'll see the typical artefacts you get with mapping a rectangle to a sphere. Though I've not tried, you could probably modify the code (use, say, "view page source" to read it, or save a local copy and edit) to load whatever texture your heart desired.