Tl; dr: Some sea-borne people like the Vikings could and did, with the proper equipped type of ships, but it was not so norm until the end of the Middle Ages, at least in Europe (I assume OP focuses on there on ground of the use of periodization term).

(Added): In the end of the Middle Ages, at least the English and Basque fishermen also sailed far and wide in the open ocean in the Atlantic up to Iceland for deep-sea(?) commercial fishery in the beginning of the 15th century (as I noted in What knowledge did mainland Europeans have of Iceland, Greenland and Vinland during the Middle Ages?, then the offshore of Newfoundland around 1500.

There will always more to be said, but I wrote some answers relevant to OP's question in the following thread:

• How Dangerous was Sailing in the Ancient and Medieval Worlds?
• When and how did straits like those of Gibraltar, Bosporus and Denmark become an unpassable obstacle for ships that were rejected passage?: It means that they didn't always have to sail in open sea route unless the route indeed had a favorable sea strait or the seasonal wind like monsoon, since the coastal authority often had difficulty in enforcing the traffic control even on the strait until the end of the Middle Ages.

Yes, certainly -- Polynesian navigators had settled well into the central Pacific by 3000 years ago, so about 1000 BCE, and had likely reached South America before Columbus came from the other direction. There's plenty on Pacific Islander expansion and navigation in our FAQ.