Thanks for the reply!
Do you think that part of the problem might be in our English use of the terms love and hate? We tend to use them as mutual exclusives. I don’t think the Scripture uses them that way. For example, we are told to love our enemies (Matt. 5:44) and our brother (1 Jn. 3:14) and also to hate our enemies (Ps. 139:21) and our brother (Lk. 14:26). To my knowledge, no one is going to say that Christians have to either love or hate. Why then should we say that God cannot both love and hate at the same time (see Calvin’s Institutes 2.16.4)?
I think a good working definition for hate (Heb. sane, Gk. miseo) is “to reject.” This seems to harmonize well as we reject anyone that comes between us and God (Lk. 14) and God predetermined to eternally reject Esau (Rm. 9). It does not necessarily mean that God has no kind of love (agape) for the reprobate.
1. No, I do not think the problem is the English use of the terms 'love' and 'hate'. But rather, I think it is the predominance of bad theology, aka: idolatry that presupposes that God loves everyone, albeit differently. The Scriptures certainly do use the terms in the way I have expressed. This type of thinking also effects how many understand the atonement. It is not uncommon to read/hear people assert that "in some way" Christ died for everyone. But that's another topic.

2.
Institutes of the Christian Religion 2:XVI:4 when read in CONTEXT is speaking about God's eternal love for the elect, those who God determined to redeem in Christ. With them, Calvin writes 2:XVI:3 God has a hatred and wrath due to the sinfulness that they possess by nature (cf. Eph 2:3; et al). However, since it was God's eternal good pleasure to save a remnant of the fallen human race, it was antecedently, absolutely necessary that the Son take upon human flesh and suffer God's infinite wrath in their room and stead. Nothing in Scripture speaks of God's 'love' for the reprobate. However, as I wrote previously, God does express His goodness and kindness (philanthropy) toward all men indiscriminately.
3. Methinks that is YOUR "working definition" for the word 'hate', but unfortunately it is without warrant. Consulting
A Greek Lexicon of the New Testament (Arndt, Bauer & Gingrich; 524:D), there is no mention whatsoever of 'rejection' in regard to the use of the word
misew. The usage of that word is restricted to "hate, persecute in hatred, detest, and abhor". This is why I provided the references in my previous post and especially, Malachi 1:1-3, from which Paul quotes in Romans 9:11-13, that vividly describes God's hatred for Esau and his descendents. In that passage, the issue was Israel's contention that God did not love them due to the suffering they were experienced at God's hand. And thus, God through the prophet explains that they being Jacob's descendants received manifold favor and blessings vs. the wrath and hatred of God for those from Esau.
4. Hyper-Calvinists insist that not only does God have an eternal hatred of the reprobate (true), but that EVERYTHING that God does for them in this life stems from that hatred (false). In short, they reject any form of kindness or benevolence toward them whatsoever. That is NOT my view.

5. Lastly, the Scriptures teach that true believers are to emulate God's holiness and show kindness to unbelievers. But we are likewise to 'hate' all enemies of God.
Psalms 139:19-22 (ASV) "Surely thou wilt slay the wicked, O God: Depart from me therefore, ye bloodthirsty men. For they speak against thee wickedly, And thine enemies take [thy name] in vain. Do not I hate them, O Jehovah, that hate thee? And am not I grieved with those that rise up against thee? I hate them with perfect hatred: They are become mine enemies."
I have found that few are even privy to this passage of Scripture and usually get upset when it is brought to their attention due to their presuppostion that only 'love' is to be felt and expressed toward all men because 'God is love' and loves everyone.