It does seems contrary that a God who is love can also hate. We are created with the capacity to both love and hate; it is part of our very being created in the image of God. The fact that we are all stained with sin does not negate the fact that the ability to love and hate is part of the image of God that was created within us all. So, if it's not a contradiction for a human being to be able to love and hate, then much more so would it not be a contradiction for God to be able to love and hate.
When the Bible does speak of God hating, the object of God’s hatred is usually sin and wickedness. Among the things God hates are idolatry (Deuteronomy 12:31; 16:22) and those who do evil (Psalm 5:4-6; 11:5). Proverbs 6:16-19 outlines seven things the Lord hates: pride, lying, murder, evil plots, those who love evil, false witness, and troublemakers. Notice that this these things do not include just things that God hates; it includes people as well.
God hates these things because they are contrary to His nature—God’s nature of being holy, pure, and righteous. David writes in Psalm 5:4, “For thou art not a God that has pleasure in wickedness; neither shall evil dwell with thee.” God is holy and hates sin. If He did not hate sin, He would not be holy. God is love, but He is also wrath, justice, and vengeance. But His wrath is a holy wrath and His justice and vengeance are holy as well. God’s love is holy. Therefore, He cannot "love everyone all the time no matter what they do," as some like to claim. God loves righteousness and holiness and He hates sin and evil.
So even though God hates sin and loves holiness, He loves us because we have the righteousness of Christ who became sin for us on the cross (2 Corinthians 5:21). He poured out His wrath and vengeance against sin on His Son, so that He could pour out His mercy and love on us. But without that sacrifice done for us, His wrath and hatred remain on us because He hates our sin. The Bible never says He hates the sin, but loves the sinner. Psalm 7:11 says, "God judgeth the righteous, and God is angry with the wicked every day."
Even as a Christian, questions put forth to us, like yours above, are definitely worth a deserving answer. It's hard to answer though when even to me it's hard to understand. I have to remember that God's ways and thoughts are higher and deeper than ours. We don't see the whole inside situation to know exactly why He seems contrary to His own words.
One other thing about God being jealous, yet we are not to envy, is that God's jealousy is only for what should be given to Him (our belief in Him, praises, worship, talents, etc.) where our envy is usually toward wanting things other people have that we don't. I do believe that an example of a jealousy that could be compared to the kind God would not be upset over, is when another man looks upon someone's wife with envy, and the husband gets that jealous feeling. He is only wanting his wife's love and affection and the other man should go find his own wife to love.
I realize this doesn't completely address the question but given some extra time, I would like to search for some more answers. Hope you are having a nice weekend! I'll try to get back soon.