So, should you include racism in your story? You have a lot of groups in conflict, and they're probably going to have a lot of hostility toward each other.
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If you accept the definition of racism as a belief in innate qualities which make different races or groups inferior or superior, I wouldn't bother with it unless that belief is a specific part of the plot or a character arc.
If you consider racism to be the disliking or hating a of group for whatever reason, then you're going to have to include it, because if you don't your story will probably be a painfully unrealistic rose-tinted utopia that could very well insult your reader's intelligence.
It is completely reasonable for characters to have negative opinions about groups whose members have harmed them in some way, even if these opinions are inaccurate generalizations.
Break free of the thought restrictions that others are trying to force onto you with their ideologies and social demands, conventions and norms that they enforce with spastic rage and unwarranted shaming.
Then, use your own logic and intellect to determine how the people in your story world would reasonably think, feel and act according to what they've been taught and what they've experienced.
Then, ask yourself: What is racism?
The belief that all members of each race possess characteristics, abilities, or qualities specific to that race, especially so as to distinguish it as inferior or superior to another race or races.
This next quote is from the same definition, but it better defines actions motivated by racism.
Prejudice, discrimination, or antagonism directed against someone of a different race based on the belief that one's own race is superior.
‘a programme to combat racism’
Bear in mind that racism is a belief.
Now consider prejudice.
An unfavorable opinion or feeling formed beforehand or without knowledge, thought, or reason.
Any preconceived opinion or feeling, either favorable or unfavorable.
unreasonable feelings, opinions, or attitudes, especially of a hostile nature, regarding an ethnic, racial, social, or religious group.
Racism is a belief. Prejudice is an action. Prejudice is an internal mental action, a thought, but it is still an action. Discrimination is an external action where someone discriminates against another person.
Now that we have some definitions, here's an example:
"Elves are naturally excellent archers."
This is a racist statement because it attributes a skill to a Elf because of his biological heritage, his race.
"Elves are excellent archers."
This is not a racist statement, because it doesn't state that Elves are excellent archers because of their race. The statement is a generalization, because somewhere there's an Elf who's an incompetent archer, but the statement is not racist.
"Elves are excellent archers because they're trained from birth."
Again, this is not a racist statement, because it states that Elves are excellent archers because of their training, not their race.
This belief that Elves are excellent archers, whether it is racist or non-racist, can cause racial discrimination:
"Nay, Willfire, you cannot join my elite troupe of archers, for you are not an Elf."
The person denying you admittance to the archery troupe is 1) prejudging you, because he decided you aren't as good an archer as an Elf without even seeing you shoot, and 2) discriminating against you because of your race, because you are not an Elf.
The notion that 'Elves are excellent archers' is also a stereotype.
: something conforming to a fixed or general pattern especially : a standardized mental picture that is held in common by members of a group and that represents an oversimplified opinion, prejudiced attitude, or uncritical judgment.
So we've nailed down definitions (to paraphrase):
Racism: The belief that races have innate qualities which make races inferior or superior to each other.
Prejudice: To form an opinion before having the facts or a decent amount of information that would enable one to draw a reasonable conclusion.
Discrimination: To treat someone differently because of a reason not related to their merit or qualities relevant to the issue.
Stereotype: A standardized oversimplified opinion or uncritical judgement (a judgement without looking at the specific facts).
Now let's take it to your story.
You have a lot of groups in your setting: Native American, Viking, English, Irish and Greek. These groups are probably going to be in conflict, and therefore they may not like each other very much.
Consider what the characters in your story have learned, both from what they've been taught and from what they've experienced. Look at the world through the perspective of each character.
A character who has lived through a Viking raid will probably think Vikings are bloodthirsty murderers, no better than ravening wolves, and so Vikings should be put down like wolves.
A character who was pushed off of his land will hate and despise the group that did it, and form unreasonable generalizations about them, like they're all heartless greedy exploiters.
A character whose home is destroyed and ends up as a vagabond falling in with a group he previously feared can discover that while that group can generally have fierce warriors, they also display love, affection and honor. The character can come to understand there is good and bad in that group, even while struggling with his grief and desire for vengeance. Or the character can reject it, and stick with the conviction that they're all killers deserving of being killed themselves.
Note that the character's intellectual conclusions and opinions often serve deeper motivations. The character wants revenge more than he cares about facts, so he sticks with conclusions which make it easier for him to seek revenge. See what I mean?