Tiene comes from tener (“to have”) and attributes a characteristic to the road: “the road has many curves.”
By contrast, hay expresses existence: “there are many curves on the road.” Both are grammatically correct, but
• La carretera cerca del río tiene muchas curvas. focuses on the road’s feature.
• Hay muchas curvas en la carretera cerca del río. just states that curves exist there.
Yes, depending on context:
• Ruta often means “route” or a designated path—common in Latin America.
• Camino refers to a smaller road or even a trail.
• Autopista is a high-speed highway (often tolled or divided).
Use carretera for a standard paved intercity road; choose the alternative if it better fits the type of road you have in mind.