The Polynesians did not-- though there are a few oral traditions, specifically in Hawaii that may refer to fishermen or other travelers from the Ryukyus (Okinawa) or Japan that ended up stranded in the islands. These stories definitely aren't outside the realm of possibility-- but aside from some intriguing straight cut stone work, which has some similarity to that of Ryukyu islanders, there isn't a material record to complement the oral traditions.

That said, if we look further back to the ancestors of the Polynesians-- the Austronesians or more specifically the Lapita Peoples, then yes, there was definitely some contact, in fact the Polynesians likely descend from a coastal people in Southern China that migrated out to Taiwan, carrying with them domesticated animals like the chicken or the pig. While settling the island of Taiwan additional crops were acquired or domesticated such as taro, breadfruit, sugar cane, turmeric. Though they have largely been supplemented by Chinese migrants in recent centuries, there are still some of these original communities left on Taiwan who speak Austronesian languages (distantly related to the Polynesian languages at this point).

There was also extensive contact with South East Asia-- the Austronesians who left Taiwan settled first in the Philippines before beginning to branch off into a Western group that settled in mostly Island Southeast Asia (and eventually Madagascar) and an Eastern Group that pushed out into Near Oceania. The division between the two groups isn't really that neat-- there was intermittent contact between peoples-- so the division is a bit arbitrary in some places.

So yes, in the distant past there was contact, but that was well before there was a unique cultural group called Polynesians. Perhaps worth mentioning that other Pacific Peoples have had much longer contact with East Asia. The Philippines is a great example of this-- with so much contact and exchange that many scholars completely carve it out of the Pacific and include it as part of Asia. But thats the nature of arbitrary boundaries. Further, island like Guam or Palau maintained occasional contact with East Asia (or at least with the Philippines)-- we can see this in some pottery styles, like the Marianas Red, but especially the introduction of Rice culture to Guam.

• Patrick Vinton Kirch, On the Road of the Winds; An Archeological History of the Pacific Islands Before European Contact (California: University of California Press, 2000).

Probably still the best overview of archeology in the Pacific-- though looking at the date its probably about time someone else write a textbook-ish overview of the field. It has a good overview of the Out-of-Taiwan and early settlement of the Pacific, during which there was more contact with East Asia.

• Matsuda, Matt. Pacific Worlds: A History of Seas, Peoples, and Cultures. Cambridge: Cambridge University Press, 2012.

Matsuda's book is probably the best place to get an understanding of how Island Southeast Asia and the Pacific Islands are connected, his book includes both regions (and some East Asia/Americas) to give a much broader picture of the ways in which all of these places have been connected and influenced one another in the past.

Not really. This is a question best answered by prehistorians (as none of the Polynesian island people had written records and trade would be best attested to by artifacts).

Western Polynesia no longer had pottery. Eastern Polynesia made their own, the practice being brought with them when they left SE Asia (Formosa/Taiwan).

See A. Smith's book, An Archaeology of West Polynesia Prehistory (2002). Or "Rethinking Polynesian Origins," in Archaeology (2010, Authors: Addison and Matisoo-Smith).

Each island has been studied separately (almost every inhabited island has had more than one archaeologist show up). Each study shows a lack of trade amongst even the Polynesian islands, and no trade with mainland Asia. If there's one closer island out there that has had recent study, I can't find evidence of it.

For Fiji (which is a great representative example and much closer to Asia than Hawaii), see DV Burley "The archaeology of Fiji/West Polynesia in the post-Lapita period," in Archaeology (2003).

For the Cook Islands, see R. Walter "Anai'o..." in New Zealand Archaeological Association, 1998.

For a literature review that really goes into depth and might uncover some exceptions to the rule of "no trade," (there are 500 separate archaeological articles summarized here), see

Kirch, Patrick V., and Jennifer G. Kahn. "Advances in Polynesian prehistory: a review and assessment of the past decade (1993–2004)." Journal of Archaeological Research 15.3 (2007): 191-238.

It was a busy decade and I haven't yet read this, but perhaps someone who is a Polynesian archaeologist will come along and summarize. If there were evidence of extensive trade, then the newer editions of books on prehistory (like all of Fagan's work) would have revised by now.