Is it permissible for a husband to prevent his wife from visiting her family, or going out for her needs, or shopping on her way to work, knowing that she stipulated this in the marriage contract and he agreed to it? If he refuses, is she permitted to go out without his knowledge?

The mentioned conditions are valid according to the Hanbali madhhab (definition), which is the practiced one. The validity of these conditions means that the wife has the right to annul the marriage if the husband violates the condition. As for the obligation to fulfill the condition, what is well-known among the Hanbalis is that it is mustahabb (recommended) but not wajib (definition) (obligatory), while Ibn Taymiyyah held it to be obligatory. If the condition is valid, the wife may go out for what was stipulated without the husband's knowledge, unless the husband sees that her going out would lead to harm (mafsadah), in which case he may prevent her. If the husband permits his wife to go out for work, there is no objection to her buying something on her way without his knowledge, unless he forbids her from doing so.