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If as a manager you had a situation of conflict/friction between some team members, setting aside their responsibility to collaborate in improving the situation, do you also look deep into the root cause, hearing each person story and asking other members as well to really understand what's going on?

Or as long as the team members find a way to work around the friction that's adequate?

As a manager, you are essentially responsible for your employee's performance - or, at least, keeping them positioned to perform well (the actual performance is up to the employee).

When there is conflict that is severe enough that it's affecting performance, that becomes part of your responsibility, the same as any other issue impacting performance - poor equipment, unrealistic deadlines, unclear requirements, etc.

That said, there is a fine line between taking responsibility for the conflict yourself, versus coaching employees such that they're able to fix the conflict or perform despite the conflict. In some cases, conflict will be up to you to resolve, or at least mediate: employee A wants to use tool X, but employee B wants to use tool Y. It's appropriate for you, as a manager, to help resolve situations like that, and ultimately have the final vote in which tool is used. The issue may present as a "personality conflict" where two team members are simply not getting along, the fact that it's due to tool selection may not be apparent unless you spend some time understanding context.

In other cases, there may be conflicts that are best handled by an EAP, HR, or other entity - personal, non-work related problems, prejudice, etc. The important thing though is that you won't know unless you do at least some level of investigation: in other words, you shouldn't necessarily assume that you need to investigate every issue to it's very root cause, but it's equally as poor an approach to not investigate any issues and assume that speaking sternly at employees will lead to them resolving things on their own without any outside help.

Absolutely not, if an issue like this becomes more than petty and work priorities are being threatened, I sternly warn both individually, trying as much as I can to stay out of the details. This usually fixes the problem as they know my next step will be to HR, and when HR gets a hold of something like this, it usually doesn't go well for either of them.

I treat my people like adults and let them settle things. I honestly don't care about causes, actions are what matters. Besides, You never get the real truth

Now, if actions are out of line, it's time to step in, other than that, no.

We're not psychotherapists and we're not trained to be, and thus unqualified to even attempt to get into "root causes", if there even is such a thing. Personally I think it's just a fancy way of saying "excuses".

Either people behave themselves or they don't. If you can't get along with your coworkers, you don't belong at that job. A manager is not a father-confessor, a psychotherapist, a high school principal, or a cop. It's not a manager's job to try to plumb the depths of why johnny is insecure and how it goes all the way back to an incident with his puppy back in fifth grade.

Now, if a person is having obvious problems, I would direct them to get help, but it's way above my paygrade to try to do it myself.

I’m going to take a slightly different direction while agreeing with @dwizum.

Your specific question asked “as a manager”. Approving timesheets and PTO, assigning tasks and being a conduit between resources is management. For some roles that’s all that’s needed or maybe all that you want to be. But if you want to lead, not just manage, then you need to be willing to dig deeper. In order to lead, you need to demonstrate interest in them and investment in the common goal you want them to buy into. Forwarding the issue to HR is within your scope, but does little to show you’re personally committed to their success.