Sinfulness and righteousness are not determined merely by actions. They are also determined by our thoughts and intents.
Prov. 12:5 provides an excellent example of this:
The thoughts of the righteous are just,
But the counsels of the wicked are deceitful.
Jesus also taught about this in his Sermon on the Mount, such as in
You have heard that the ancients were told, "You shall not commit murder" and "Whoever commits murder shall be "liable to the court." But I say to you that everyone who is angry with his brother shall be guilty before the court.
Romans 14:141, when discussing whether or not Christians must follow the Mosaic Law, Paul says that if you believe something is unclean, then it is unclean for you:
I know and am convinced in the Lord Jesus that nothing is unclean in itself; but to him who thinks anything to be unclean, to him it is unclean.
Eating something which is unclean is a sin. In other words, Paul is saying that if you believe that eating a particular food is sinful, and you do it anyway, then, for you, it is sinful. God does not judge us only by our actions, but also by our thoughts, emotions, and intentions.
Just as sin involves both intentions and deeds, love also involves more than just deeds. Love is also a state of mind. If you intend and desire to love others, even when you do not currently have the opportunity to do so, you are still following the law, as repeated by Christ, in
You shall love the Lord your God with all your heart, and with all your soul, and with all your mind.
Notice that the law was not to "love God in all that you do", but rather to love Him in your heart, soul, and mind. God is concerned with the matters of the heart. God wants people to love him with their thoughts and intentions, not just with their outward actions. To love, only by your actions, is to be a hypocrite (Paul warned against loving hypocritically in
Rom. 12:9). That is why God was more interested in the circumcision of the heart than He was in the outward physical circumcision. For instance, in
Deut 30:6 the the Bible says:
Moreover the Lord your God will circumcise your heart, and the heart of your descendants, to love the Lord your God with all your heart and with all your soul, so that you may live.
Romans 5:5 Paul describes how, as Christians our hearts have been changed:
...the love of God has been poured out within our hearts through the Holy Spirit who was given to us...
2 Tim. 1:7, Paul also talks about a "spirit" of love:
For God has not given us a spirit of timidity, but of power and love and discipline.
Having a spirit of love, that was given to you, does not require that an object for that love was present at the moment the spirit was given. A heart filled with love does not cease to be filled with love when the Christian leaves the company of others. A circumcised heart will not regrow its metaphorical foreskin when it is lonely. If it were not possible to remain loving towards others in spirit, even when others are not around, how could God judge our hearts and our minds?
When John stated that "God is Love", he was not referring to any specific actions. He was not describing the love that God has for any individuals in particular. He was describing God's unchangeable nature. As the verses above demonstrate, it is possible to have a nature of love without having any object to receive your loving acts.
From a purely logical basis, since sin is not eternal, the same could be asked of God's mercy and long-suffering. How can you explain that an unchanging God was merciful before sin existed? I would argue that mercy and love are part of God's unchanging nature and someone's nature doesn't change depending on their circumstances.
For Trinitarians, God's love compliments their theology perfectly, as you described. For Unitarians, however, God's love does not contradict their theology. God's nature doesn't prove that the Godhead necessarily includes multiple persons. On the contrary, it actually provides an explanation for why God may have created us. If His nature is love, it follows that He would create us so that He would have someone else with whom He could share His great love. That is not to say that God needed us, in any way, but rather that the venture would appeal to Him, if you will.
Another, slightly less convincing, argument is that time, itself, is part of the creation. Following that line of thinking, from God's perspective, there is no such thing as "before" or "after" creation because "before" and "after" are part of creation. Saying that there was a "time" when God was alone is nonsensical when you consider that He is outside of time entirely.