Is saying to my friend, "I am going to perform ablution," without specifying prayer or removing makeup before ablution, considered an intention for ablution or Dhuhr prayer?

It is not a condition to specify the prayer in the intention for ablution. Rather, it suffices to intend to remove ritual impurity or to make permissible prayer, circumambulation (tawaf), and touching the Mus'haf. If you intend for the prayer for which you will perform ablution, this is not sufficient according to many scholars for the intention of the prayer itself, because they stipulate that the intention for prayer must be connected to the takbirat al-ihram (opening takbir) or precede it by a short time. However, some have permitted a separation as long as the intention is not diverted to something else.