Well, all right — this is just the case of 'a strong dragon cannot crush the snake that owns the ground.'

也罷: 罷 = ba, the light word that softens a sentence into "oh, all right then" -- and a different word from the bà le of 罷了 back at the mock concession, though the two share a character. Then the proverb, which the edition marks as a quotation: 強龍不壓地頭蛇, "a strong dragon does not crush the local snake." It is still in daily use, and it means that however powerful you are, on somebody else's ground you defer to whoever knows the terrain. Wukong is calling himself the dragon and the immortal a snake, and dressing it as good manners.

You go first, sir; but we must first make things clear before the sovereign.”

先生先去 "you go first, sir." 對君前講開 "to set it out plainly in front of the ruler."

The immortal said, “Make what clear?”

講甚麼: 甚 = shén, "what."

Pilgrim said, “If you and I both go up to the altar to pray for rain, how is one to know whether the rain is yours or mine?

The question that springs the trap, and it is disarmingly practical. 都上壇: 都 = dōu, both of us going up together. 行者: 行 = xíng.

There'd be no telling whose the achievement was.”

不見是誰的功績了: the falling tone of 見 pulls 不 to bú. 功績 "merit, achievement" -- and credit is what the argument is really about.