Mi primo está de visita en Madrid esta semana.

Estar de visita is an idiomatic expression meaning “to be visiting” as in “to be here as a guest.” It uses the verb estar plus the noun visita.
By contrast, visitar is the action “to visit” (a place or a person).
• Mi primo está de visita en Madrid = “My cousin is here visiting Madrid” (focus on his current status).
• Mi primo visita Madrid = “My cousin visits Madrid” (focus on the action of visiting).

The de is part of the fixed phrase estár de + [sustantivo] which indicates a temporary state or role. Similar examples:
• estoy de vacaciones (I’m on vacation)
• está de acuerdo (he/she agrees)
Without de, the meaning changes or the sentence becomes ungrammatical.

In Spanish, most city names are used without an article:
• en Madrid, en Buenos Aires, en Lima
You would only add an article if the city’s name traditionally includes it (e.g. el Cairo, la Habana) or for stylistic reasons, but not el Madrid.

Yes. Spanish has flexible word order for emphasis:
• Esta semana mi primo está de visita en Madrid. (emphasizes this week)
• Mi primo está de visita en Madrid esta semana. (neutral)
Both are correct; you choose based on what you want to highlight.

You can, but there’s a nuance:
• Mi primo visita Madrid esta semana uses visitar (the action of visiting). It’s direct and neutral.
• Mi primo está de visita en Madrid esta semana highlights his temporary state of being a guest.
Use whichever fits your emphasis: action vs. status.