To what extent were Polynesians in contact with one-another before European contact?

"Was it a continuous and extensive trade network with political interactions like in the ancient Mediterranean?"

No. Unlike the Mediterranean, trade is much more marginal in Polynesia. The problem is that all of the islands pretty much all had the same resources. Now, within the same island chain, there was potential for specialisation in comparative advantages. One notable example is the trade of moa meat from South Island to the North Island within New Zealand - there was bird meat in the north as well, but the relative abundance in the south meant short-distance trade made sense.

Beyond nearby islands within the same island group, the vast distances rapidly makes most trading not worth the trouble.

Although inter-island canoes did ply between the islands of the group and to a few islands outside it, trade was only a small part of the Tahitian economy . . . trade between the volcanic islands which contained the bulk of the population was probably limited to unique specialties, such as a special type of red feathers for the adornment of sacred loin-clothes worn by ruling chiefs, for each high island contained most, if not all, the range of resources available in the group as a whole.

_{Finney, Ben R. Polynesians Peasants and Proletarians. Schenkman Publishing, 1973.}

However, as the example above hints, the main exception is religious or ceremonial items.

"Did such a trade network exist only in the core areas, such as in the the Tu'i Tonga empire, with outlying islands being isolated?"

In fact the Tu'i Tonga trade network is an example of such trades in ceremonial goods.

It appears that the pre—eminent Tongan context for the use of Fijian and Samoan trade goods was, and is, on ceremonial occasions and especially weddings, funerals, and various kinds of state and religious celebrations. At funeral presentations Samoan fine mats are the most important material object - a number of them being necessary as a cover for the dead and as gifts.

_{Kaeppler, Adrienne L. "Exchange patterns in goods and spouses: Fiji, Tonga and Samoa." The Australian Journal of Anthropology 11.3 (1978): 246-252.}

This went hand in hand with the other major reason for transoceanic Polynesian contact - marriage.

In Tongan society, a woman and her children were traditionally of higher rank than her brother. If the highest ranekd sister of the Tu'i Tonga married a tongan, her son, being of superior rank, could pose a threat to the ruler's political position. By marrying a prominent Fijian, the Tu'i Tonga Fefine protected the Tu'i Tonga's status, since her offspring were considered to belong to the Fijian line, the Fale Fisi, and thus ineligible for Tongan kingship. For the hau or active ruler, a marriage with a Tongan woman might produce children of diminished rank, a problem the hau and other Tongan nobles often avoided by marrying high-ranked Samoan women. From the Tongan standpoint, Samoa served as a wife-giver and Fiji as a "husband-giver".

_{Hommon, Robert J. The Ancient Hawaiian State: Origins of a Political Society. Oxford University Press, 2013.}

In this way, dynastic concerns served to maintain a link between the disparate edges of the so-called "Tu'i Tonga empire". The resulting familial ties were a major reason for long distance voyages between far apart islands. Early Polynesian settlers of both Hawaii and New Zealand made return trips for genealogical visits.

Unlike shorter distance examples like the Tui'tongans, however, Hawaii and New Zealand were too far away to keep up regular exchanges. Voyagers grew more and more infrequent until they ceased altogether when family ties died out, literally.

"Were interactions rather local, to neighboring islands only, with a few rare examples of daring explorers visiting far-away lands?"

As seen in the case of the Tu'i Tonga example, the islands were not actually "neighbouring". However, in general most voyages were indeed only regional, at least relative to the vast expanses of the Pacific. This does not mean only "daring explorers" range far, though. Beyond the initial discovery, detailed knowledge of how to reach the islands of Polynesia were often passed down generation to generation.

One example was Tupaia, known to the West for having boarded the HMS Endeavour during Captain James Cook's voyage to New Zealand. Even though he had only ever been to a few personally, mostly nearby ones, Tupaia knew of some 130 islands across Polynesia, including Samoa, Tonga, and Fiji on the eastern end of Polynesia, and Marquesas to the east. Moreover, he remembered from his forebears detailed instructions on how to get to the islands.

He was not, however, aware of Hawaii or New Zealand.

"Did this change over time with periods of extensive trade and integration and periods of the breakdown of inter-island exchange?"

The curious thing about the aforementioned gap in Tupaia's knowledge is that Hawaii was reputedly settled by Marquesans and Tahitians. This indicate that Tahiti's knowledge of the further islands did indeed die out after the voyages ceased.

In contrast, kinship ties to Taihiti was preserved in Hawaiian memory.

It is thus on account of her being the mother of chiefs, both here [Hawaii] and in Tahiti, that she is called Papa Nui Hanau Moku. She is said to have been a comely, handsome woman, very fair and almost white. Papa is said to have travelled eight times between Tahiti and Hawaii, and died in a place called Waieri, in Tahiti, during the time of Nanakehili, the fifth in descent from her and Wakea.

_{Cartwright, Bruce. "The Legend of Hawaii-loa." The Journal of the Polynesian Society 38.2 (150 (1929): 105-121.}

"Was ocean-seafaring done by only a small group (a caste? a tribe?) of the wider population?"

It depends on how far you mean. Obviously, only vey specially trained navigators, as Tupaia was, could have known how to reach islands hundreds and even a thousand kilometers away. They could not have seen where they were going, and had to rely on specialist knowledge of navigation by sun and the stars, as well as wave and wind patterns.

On the other hand, the ability to kayak between nearby islands within a local island group was not remarkable.