Will the graph of a system of equations always intersect at exactly 2 points?

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No, only sometimes.It can intersect at 1 point or none at all (i.e. when the lines are parallel).If you're talking about 2-variable system of equations, the graph can never have 2 or 3 solutions. Only 0 (when 2 lines are parallel), 1 (2 lines intersecting) or an infinite number of solutions (same line; overlapping). 

no the lines can have the same slope and be parrallel to one another. or they can intersect in one point. if the grapgh is 3-d then they can be skew lines