Network Analyst solves routes using the network dataset. The network dataset provides the cost attribute that it minimizes on and a set of restrictions that can turn on/off places that can be traveled on, You can also import or set up temporary barriers.
That said, the cost attribute can be made from a combination of factors/fields in the input data. But in the end, there is only one cost attribute you can pick at a time. The route solver will not try to minimize more than one cost. So if you can combine the multiple fields into one cost then yes Network Analyst to solve on it. And you can add multiple cost attributes to a network dataset from different fields and pick any one of them to solve on.
As far as selections, what role are are they playing? Do you have a set of points to visit and you make a selection to visit a subset? If that is the case, when you Load or Add Locations to the stops, it does honor selections. The Load Locations has a check box for "Only load selected rows" and it will be Checked on by default if you have a selection but you can un-check it. If you are trying to select edges to "restrict" the travel to, then you will have to use the restriction model as I mentioned above to model these selections.
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