The philosophy, virtues, and views of the peoples in the Elder Wood are geared so that these people come into conflict all the time, but the reasons that they come into conflict are a little different from historical or usual fantastic ones. Here are a few notes on what is and isn't usual:
Pride
There are several stories among the tales about prideful and vain people; these exist because this is a common problem in the setting. By modern standards, the people of the elder wood are trained to be deeply self-confident, and this flips over into arrogant behaviour fairly often.
Lies And Insults
The common social reaction to feeling lied to or insulted in the elder wood is violence. This violence is often somewhat restrained; you don't hit someone when you're both guests - you go outside, maybe set terms like using wooden blades, and then go to it. A musketeering duellist would be right at home with this, once they got used to the informality of it.
Feuding
The belief that some piece of violence was too much, of course, incurs more violence, which leads to... Yeah. Like that. Kings and queens and elders may well need to call a trial just to put an end to this kind of thing. Some feuds carry on to the point of becoming ancestral grudges, which brings us to...
Racism
Anyone with any known Faerie ancestors is liable to face some level of what is functionally racism; a little mistrust from everyone, and more trouble than that from a few. This is a whole-culture ancestral grudge following from the Balachin invasion and occupation. Those affected can also include anyone with magical powers, under the belief that these are connected, but only in fairly extreme cases. Notably, racism does not travel along lines of skin colour; people as light as alder-wood, middling as oak-wood, and dark as hazel-wood get along as fine as anyone else, and anyone asserting racism on those lines would be viewed fairly universally as a vicious idiot.
Dominance
Attempts by people to exert abusive dominance over other people, including successful attempts, do occur. These take place most often over lines of existing structures, martial prowess, wealth, sex and gender, and magic. These actions are considered immoral in this order - amplifying existing structures is troubling but often accepted, broad social dominance by warriors (outside of the "safe" structure of champions, among the Saesin) is immoral, building a class hierarchy on wealth is worse, debasing one sex or the other worse still, and building a system where magicians universally hold the rule (again, outside the "safe" structure of coven territory among the Saesin) is considered entirely evil. As an aside here, most situations of dominance along gender lines are likely not worth the kind of effort it would require to do them well rather than as a cheap and lazy grab for motivation, and are likely best kept to the deep background rather than as something characters are expected to take action around.