No. Most degree programs for international students are offered entirely in English. However, learning Danish can be beneficial for internships, part-time jobs, and social integration.
Yes, international graduates can apply for a post-study job-seeking visa which allows them to stay in Denmark for up to 3 years to find employment. Once employed, they can apply for a work permit and eventually permanent residency.
Yes. You can apply for family reunification or dependent visa, but you must prove sufficient financial resources and accommodation. Processing times and conditions vary. A spouse can work for the full time once he/she arrives and one can claim for the educational scholarship for their child as well.