Before attempting to answer, please allow me to give you some background on the question. Only a year ago, I was a hyper-Calvinist. I know that the use of the term is somewhat subjective, so to be clear:

I denied the offer of the gospel
I denied common grace
I described God’s benevolence as “fattening the reprobate up for the slaughter
I denied any kind of love of God for the reprobate
I believed that evangelism meant trying to persuade people of election and reprobation.
I spoke of God commanding everyone universally to repent but never promised remission of sins

I think by the definition of most, I was a hyper-Calvinist, but like most hyper-Calvinists, I would never have called myself that. Honestly, it was my current pastor that helped me see it as well as a multitude of reading I did from Calvin and some others. I have also in the past year spent very much time studying the Three Forms of Unity and am in 100% agreement with these confessions as accurate interpretations of Scripture.

In researching hyper-Calvinism on the-highway, I came across postings that distinguished between God’s love and His benevolence, and I simply wanted to have a discussion to see how such a distinction is understood. Of course I have an opinion on the matter, but since I am fallible and subject to error, I am willing to be wrong if I see sufficient biblical proof that there is a distinction.

At this point, I’m going to try to briefly answer the questions that were posited.

I do not believe that God hates the sin of the reprobate but loves the sinner as if He has no hate for the person of the reprobate. The Scripture says He hates the person. This may be on account of the sin in which the decree of reprobation leaves the sinner, but it would be foolish to try to impose on Scripture such a division.
You have quoted many verses that speak about love for the elect and hate for the reprobate. I agree with all of these verses: God does love the elect and God does hate the reprobate. The particular love that He has for the elect results in salvation. But I think you create a false dichotomy. Please correct me if I’ve misunderstood you, but you conceded “we are likewise to 'hate' all enemies of God” and “we (believers) are to love our enemies by showing kindness to them.” Haven’t you agreed that it is in fact not a contradiction that believers simultaneously exercise both love and hate? How is it by necessity a dichotomy when it comes to God?
Furthermore, do we not love different people differently? I love my wife differently than I love my children and differently than I love my neighbor. To define love only in the context of marriage, for example, would not then do any justice to the love that I have for my children or my neighbor. However, if my neighbor sought with me the same expression of love that I have to my wife, I would reject her. In the words of Charles Hodge:
“By the love of God is sometimes meant his goodness, of which all sensitive creatures are the objects and of whose benefits they are recipients. Sometimes it means his special regard for the children of men, not only as rational creatures, but also as the offspring of Him who is the Father of the spirits of all men. Sometimes it means that peculiar, mysterious, sovereign, immeasurable love which passes knowledge, of which his own people… are the objects.” (Systematic Theology, Part 3 Ch. 8, sec 2 point 4)
Concerning providing one Apostolic sermon that proves that sinners are to be addressed with God’s love: I never stated that the unsaved were to be addressed that way and through the course of this discussion I have distinguished God’s love of benevolence from His love for the elect. I never stated that the object at stake here being able to tell the unconverted that “God loves you.” I’m not trying to “pluck any heart strings.”

I would like to provide Scripture proofs that I believe support my proposition.

My little children, let us not love in word or in tongue, but in deed and in truth. (1 Jn. 3:18)

If we are to demonstrate our love through actions, Christ being our chief example, only searching for the word “love” is inadequate since love is more clearly witnessed through its outworking.

Love suffers long and is kind… (1 Cor. 13:4)

If suffering long (longsuffering) is an outworking of love, even God’s longsuffering to the reprobate in Rom. 9:22 cannot be divorced from His love. We know that God “is kind to the unthankful and evil” (Lk. 6:35). We both agree that God is benevolent and kind to the reprobate. Hasn’t God Himself defined kindness as an outworking of love? Isn’t the thrust of Matt 5:43-48 “love all of your enemies the same way God loves all of His?” Isn’t it splitting hairs to say that when we show kindness it is love, but not when God shows kindness? Also, in the Hebrew thinking, isn’t a “son” one who represents His master?

If loving our enemies is commanded of us and in part fulfills the second table of the law, how did Christ obey and fulfill that law and satisfy God’s perfect requirements if He did not perfectly love all of His enemies? Didn’t Christ leave us a perfect example of how we are to love? Are we under obligation to believe that the rich young ruler whom Jesus loved had to be elect (Mk. 10:21) or to believe that those whom Christ asked His Father to forgive were only the elect out of the ignorant ones (Lk. 23:34)? How can we separate Christ’s compassion, kindness and benevolence from His love?

Lastly, I would like to consider Jn. 3:16 in context. I do understand that in reformed theology there is not a consensus on this passage particularly, but in my understanding it is one of the most beautiful proof-texts about God's universal love.

1. I believe the immediate context is Christ being lifted up before everyone as the serpent was lifted up by Moses (v. 14). With this allusion, we don't have the luxury to say that the serpent was only lifted up for those it healed, but it was lifted up to everyone with the promise of healing for whoever looked. However, we can certainly say that the only ones that were healed were the ones that looked. Therefore, the context seems to be Christ indiscriminately offered in the gospel.

2. If the word "world" in this context only meant the elect, the inclusion of unbelievers in the context who are condemned for their unbelief is irrelevant.

3. Further, Christ specifically is said not to condemn, since the condemnation stems from unbelief, not Christ offered (v. 18).

4. The phrase "but that the world through Him might be saved" is not speaking about the certain salvation of the elect but the promise of salvation offered in the gospel. It is indeed a universal promise which in the final analysis will only save the elect and condemn the reprobate because they did not believe.

5. The verses are neither proof-texts for election or proof-texts that everyone has the ability to believe. The ability is spoken of earlier in the chapter (being born from above) which of necessity has its genesis in God, not man. Also see Jn. 6:44 as a proof of who believes and why they believe.

I believe Calvin interpreted this passage correctly.

I hope all of this helps...