Is the wife entitled to request visiting her family and staying with them for a longer period, given that the visit does not exceed a week and her husband does not object, but his mother gets angry because of it?

There is no blame upon you for requesting to visit your family. This should be done according to necessity and the husband's capacity, based on what is mutually agreed upon. The husband is not obliged to make his wife live with his family. She has the right to request a separate dwelling or for him to leave her with her family.