How are extreme winds aloft handled and how do they affect flight?
Is there a maximum or safe winds aloft value / crosswind for say, a
  737 to fly?


Crosswind limits are in place for landing not flying. Cross wind limits have to do with how effective the control surfaces are which allows you to fly the plane in such a manner where the fuselage is aligned with the ground track. At altitude winds simply either add to your ground speed or count against it. 


  How is it reasonable to deal with wind speeds this high when they're
  other than a direct headwind or tailwind? What if it is a direct
  crosswind, how is it possible for the airplane to fly without
  significant navigation issues (or maybe safety issues?) at velocities
  that high.


Winds are not always a direct head or tail wind (for that matter they often are not) thus you simply need to figure out your course correction. You can check out this video on how to do so. Keep in mind that you are flying through the air and thus moving with it, on the ground you feel wind because you are stationary and it is moving around you. 


  In general, how does the significant wind speeds found in the upper
  altitudes affect flight (contrasted with typical general aviation
  flight)


Its all relative but in general it simply effects time in transit (ground speed). However keep in mind that the planes that fly up there are often faster so 150Kt headwind on a plane that can cruise at 550Kts at FL300 is not all that different from a 35 Knot headwind on a plane cruising at 140kts at 5000ft.