Is it permissible for a wife to go to her sister's husband's veterinary clinic to check on him and offer food and drink, and what are the regulations and duration for that? Also, would that be considered 'khalwa' (impermissible seclusion) if the clinic door is open and patients are present?

The husband of one's sister is not a mahram to a woman unless there is a reason such as suckling (رضاع). Otherwise, he is a stranger to her, and she must treat him as she would any other stranger: by not being alone with him (خلوة), by not speaking to him unnecessarily, and by covering her entire body (with a difference of opinion regarding the face and palms, the more correct view being to cover them).

If the wife wishes to visit her sister (the brother's wife) or offer assistance to her sister's husband, there is no objection if her own husband permits it, provided she adheres to Islamic guidelines: wearing proper Islamic attire, not being alone with her sister's husband (whether in his clinic or his home), and not speaking to him in a soft, alluring manner or conversing with him unnecessarily.

Being alone (خلوة) is precluded by the presence of the sister, or by the presence of people and open doors. Therefore, there is no impediment to the visit if temptation is avoided, and the woman adheres to the Islamic veil, lowers her gaze, and avoids speaking in a soft, alluring manner.