Is morality possible apart from God? Can an individual live according to Christian morals taught by Jesus, but not live according to the "greatest command" (to "love the Lord your God with all your heart and with all your soul and with all your mind.")?

In other words, is it possible to not love God, not follow God, and not believe in Jesus but still keep all the other commands?

Specifically, I'm thinking of commands like "honor one another more highly than yourself", love those who hate you, do not conform to the pattern of this world, do not take revenge, etc.

Is it possible to keep these types of high moral commands and maintain a separation from God?

Furthermore, what distinctions could be made between someone ignorant of the Bible and someone who has outright rejected God's Word?
As is common, it largely comes down to how you define morality. If (as per the question) one doesn't believe in God or follow God, then there is no reason to accept some of the things that are only declared "moral" due to being divinely stated. We all know the obvious ones... they have come up here often enough, so I won't get distracted mentioning them.

But yes: one could choose to adopt the same morals as the Bible, but I think everyone should be able to qualify the reasons for their morals; and if you don't have a divine crutch to support them, then there are a few Biblical morals that are pretty hard to justify otherwise.

Re the examples: I would wager that those things are equally challenging for people of all creeds and none, but both society and law generally cover most of them - for example, we don't act in a vigilante way. Likewise "honour one another" and "love those who hate you" generally fit very well with a Humanist outlook, which places emphasis on the rights, human value, and equality of all people.

The "do not conform to the pattern" one cited is interesting, though; if I understand your explanation (comments), then this is "don't worry about earthly things". Now, if we put aside the notion of God (as per the question), completely denying earthly things is... Stupid. The view is "this world (this wonderful, amazing, beautiful world) is all we have". If we don't take the time to enjoy it (with our friends and families etc), then we are missing all that we have. So: while I do not support overt materialism, I absolutely am going to try and enjoy this world and my time on it (but hopefully not to the exclusion of others).