Did John Calvin believe that the Son and the Holy Spirit are eternally subordinate to the Father per their "divine" roles or functions in the Godhead?
Calvin evidently thinks of the three persons (tres personae) as three subsistences, divine persons who are distinguished from one another "by an incommunicable quality" (Giles, 53) which he identifies as "subsistence." An incommunicable quality is an attribute that cannot be transferred or passed on to someone/something else. E.g., parents communicate certain traits to their offspring; unfortunately, some diseases are communicable too. However, Calvin reckons that each divine person within the triune Godhead possesses some quality that cannot be communicated by the bearer of said property. These incommunicable properties only belong to divine personae.
Subsistence in Calvin's theology assuredly does not mean "essence" (essentia). He asserts that there is a "characteristic mark" that sets the LOGOS apart from God the Father so that the Word can "be" God and simultaneously "be with God" (Jn 1:1b-c). That defining "characteristic mark," Calvin argues, is not the Son's essence but his subsistence.
It is unfortunate that Calvin does not appear to explain, at least in a thorough or analytic sense, what he means by "subsistence." In any event, Kevin Giles believes Calvin does not imply that the Son or the Holy Spirit are subordinate to the Father qua being or qua function. Giles then provides various lines of evidence for this claim on page 54 of The Trinity and Subordinationism. One such line of evidence is that Calvin reasons that both prayers and worship should be directed not only "through" the Son but "to the Son." Admittedly, Calvin does accept an "order" (taxis) in the Trinity insofar as he believes that the Deus Trinitas is structured and functions in an orderly manner. However, Calvin does not think of the Trinity in hierarchical terms and rejects any talk of one person being before or after the other person "within the Godhead" (ibid, 55). Maybe he accepts logical priority, but eschews temporality priority.
At any rate, Calvin's non-subordinationist stance appears to be demonstrated when we note his exegetical comments regarding Jn 14:28 and 1 Cor 11:3. For Calvin, 14:28 is contrasting Christ's earthly state with his "present state" and "his heavenly glory to which he was shortly to be received" (Giles, 56). Furthermore, 1 Cor 11:3 (says Calvin) appertains to Christ in the flesh since "apart from that, being of one essence with the Father, he is equal with him" (ibid). Calvin's exegesis of 1 Cor 15:24ff is also worth reading.
In short, Calvin possibly applies Paul's words in 1 Cor 15:24ff to the soteriological-mediatorial office of Christ. If this conclusion is accurate, then Calvin does not apply the Pauline account to what he would identify as Christ's eternally timeless divinity.