So is it possible to do evil without knowing it or without intending it? It is certainly possible to wrong another person without knowing or intending it. The Old Testament certainly has a category of unintentional sin. In fact, the Levitical system of sacrifices primarily provided for unintentional sin (e.g., Num. 15:22-31). We can infer from the stories of the Old Testament that there was atonement for intentional sin (e.g., when David sinned with Bathsheba), but this fact is not at all clear from the sacrificial instructions of Leviticus and Numbers themselves.
However, the New Testament shows little interest in this category. The "sins committed in ignorance" in Hebrews 9:7 relate to the time before the audience was "enlightened" (e.g., 6:4), before they experienced the Spirit and confessed Jesus as Messiah. The kind of wrongdoing that is of interest to the New Testament is not a vague violation of anything that "misses the mark" or anything short of absolute perfection. Rather, it is wronging others, particularly intentionally, and knowingly doing wrong.
Paul puts it this way, "Everything that does not come from faith is sin" (Rom. 14:23). What he means by that is when you are seriously unsure something is okay for you to do before God and you do it anyway, you are sinning. You are doing wrong. The knowledge of what you are doing is key.
Similarly, James 4:17 gives us a classic definition of what we might call "sins of omission," when you do not do something you know you should. "Anyone, then, who knows the right thing to do and fails to do it, commits sin" (NRSV). In both these cases, the key to what it means to sin or do wrong is not so much the violation of a rule but the intent to violate a rule. 
The point is that sin is overwhelmingly a question of intent and motive in the New Testament. It is not primarily some legalistic measurement against an absolute standard. One can wrong another person unintentionally or unknowingly. These kinds of incidences can be called sin, but they are important to God because of what they do to the other person rather than as a matter of great concern for God in relation to you.
Biblically, one can also unknowingly or unintentionally do things that are wrong according to God, even though they do not wrong your neighbor. It is important to recognize this category as very obscure in the New Testament. The NIV 2011 of Romans 5:13-14 says, "sin is not charged against anyone’s account where there is no law. Nevertheless, death reigned from the time of Adam to the time of Moses, even over those who did not sin by breaking a command, as did Adam." Paul seems to say that there were consequences for such sinning, even though God did not consider such individuals guilty in a moral sense.
In other words, Paul seems to support our claim even here, namely, that the question of guilt and the question of morality has to do with knowingly doing wrong. Unintentional wrongs we might do are thus not evil, even though we can consider them wrongdoing, even sin. A person could unintentionally kill someone in an incredibly tragic way. But if it was truly unintentional--and not involving any inappropriate choices on our part--then technically we would not call it evil. 
It is also clear here that the popular notion that "all sin is sin" is also unbiblical to the core. If by the idea that "all sin is sin" you mean that, before God forgives our past sins, one sin drives us to God's forgiveness just as much as any other sin, this would be correct. However, since God looks at sin primarily in terms of our intentions, it would not be true to say that God looks on all wrongdoing the same after we have believed.
As an example, you can wrong your spouse in many different ways in a marriage. For example, a husband could forget his wife's birthday. But this "wronging" of your spouse is of quite different moral significance than a husband cheating on his spouse with another woman.  The level of intention and actual wrong to the wife is of a completely different level. The relationship can easily survive missing a birthday. It may or may not survive an affair.
In the same way, both God's evaluation of wrongdoing and the consequences of wrongdoing can vary widely. Paul does not kick the arrogant and spiritually immature out of the Corinthian church. He does, however, kick out the man sleeping with his step-mother. The level of the sin correlates with the intensity of the wrong done and the level of intentionality.
The ancient model of patrons and clients gives us great insight into how such things likely worked in Paul's mind. Words like grace (charis) and gifts (charismata) from God are terms of patronage. You did not earn the graciousness of an ancient patron. The gift was always disproportionate to anything you might do for the patron. But you could solicit patronage.
Similarly, gifts of patronage technically came without payment, but this fact does not mean there were no expectations. There were informal rather than formal strings attached. At the very least, the patron expected to be praised and given honor for his or her giving. If you were ungrateful, you certainly would not continue to receive patronage.
This model corresponds well to what we find in Paul. Our right standing before God is a gift of grace from God. We solicit it with our faith but we cannot earn it. God does have certain expectations of us thereafter. Those who do not press on will not win the prize of God's "upward call" in resurrection (Phil. 3:14). Those who do not discipline their bodies to win the contest will be disqualified (1 Cor. 9:24-27). Those who insult the Spirit of grace will find themselves with no sacrifice for sins left (Heb. 10:26-31).
The notion that "all sin is sin" is thus unbiblical when applied to wrongdoing after individuals are forgiven their past sins. Sin rather becomes almost entirely a matter of one's intent to do wrong. It becomes a relational issue. Some wrongdoing, such as sins of surprise, are more a matter of neglect than high intention and have less immediate effect on one's relationship with God our divine Patron. Other sins may involve such a high level of intentionality in wrongdoing that they would revoke God's patronage all together and sever our relationship with God.
Accordingly, unintentional and unknowing wrongs do not fall under our definition of evil. Things that happen and things that are done unknowingly can have very bad consequences. But for them to fall under the "problem of evil," they must be things a moral agent intends to do on some level.
 Here it is probably worth pointing out a common misinterpretation of Romans 3:23. The New Living Translation interprets this verse to say that all have sinned by falling short of "God's glorious standard." But in fact, what Paul is saying is that all lack the glory of God, a glory God created humanity to have at the beginning (cf. Ps. 8:5) and that we hope to receive when Christ returns (e.g., Rom. 5:2; 8:18).
 In our world, drunk driving almost always does involve inappropriate choices. So if a person were to kill someone else unintentionally while driving drunk, such a wrongdoing is a moral act because it has involved a prior choice to drink and drive. It falls under Wesley's category of a "sin of surprise."
 On the other hand, a one time "sin of surprise" in forgetting your spouse's birthday can become more an more of a "high handed" sin if you do not do something about it. Sins of neglect, when not addressed, quickly become sins of intent.