Is it permissible to perform Umrah on behalf of one's wife, with the intention of improving her condition, without her knowledge, after performing Umrah for oneself? And does shaving with a number (2) clipper count as shaving or shortening?

You are not permitted to perform Umrah on behalf of your wife, given her capability and her lack of permission. You should turn to Allah Almighty, supplicate to Him, seek His help, then wisely consider the causes of the problems and strive to resolve them. There is no objection to consulting specialists. We remind you of the hadith: "Indeed, woman was created from a rib, and she will never be straight for you in any way. If you enjoy her, you will enjoy her despite her crookedness. If you try to straighten her, you will break her, and breaking her is divorcing her." And the hadith: "Let no believing man despise a believing woman; if he dislikes one characteristic in her, he will be pleased with another." Shaving (halaq) is the removal of hair with a razor or similar tool, such that no hair remains. If the clippers leave some hair, it is considered shortening (taqsir) not shaving. One who shortens with clippers on setting one is considered a shortener, not a shaver, and thus misses the three supplications for those who shave.