Learn English – turn right vs turn to the right

I checked four dictionaries and they all agreed that "alright" is an informal equivalent of "all right." However, informal doesn't mean wrong; it merely means informal. Thus, "alright" is perfectly acceptable in informal contexts.

Your rewrite does not have the same meaning as the original. Many English prepositions can be employed without an explicit complement when the complement can be inferred from the context.

When she came to the river she swam across (= across it, across the river)
These chocolates have cherries inside (= inside them, inside the chocolates)
Jack charged, and Harry was right behind (= behind him, behind Jack)

Beyond may be used this way, and is used this way in Hemingway's sentence:

It was my business to cross the bridge, explore the bridgehead beyond ... (= beyond it, beyond the bridge).

You may thus understand beyond here as a sort of 'reduced' preposition phrase acting as a locative modifier on bridgehead. The area the narrator is called upon to explore is the bridgehead itself, not some area beyond the bridgehead:

A bridgehead (or bridge-head) is the strategically important area of ground around the end of a bridge or other place of possible crossing over a body of water which at time of conflict is sought to be defended/taken over by the belligerent forces. —Wikipedia