I am afraid I don't understand what you are trying to do. What do you mean, the 'stripes of my images have lost their georeference'? It sounds like some images do not have georeferencing so you import georeferencing from other images? Do the images cover the exact same extent? They would have to have the exact same upper left pixel location and pixel size. If not, you could instead perform an Image Registration workflow to register the image to the one that has georeferencing.
Orthorectification is not necessary and only possible if your images have RPCs. It is less important in flat terrain than terrain.

Are you trying to 'fill in' areas (stripes) of missing data by mosaicking? Once the images are georeferenced, you can display them in the ENVI display and use the transparency slider to see how well they align and how well the areas are filled. If it looks good, then a mosaicking will likely be successful. There is a 'Preview' button, so you can try that too!