For example, can every character use one of their pawns to do 3 'gathering' actions in a round? All three actions would require dice rolls.
From the official rulebook: (p.6)
Each Action can be taken multiple times in the same round, except for the Threat Action, which is always a one time Action taken on a specific card, after which the card is discarded.
Only restrictions regarding "Gathering resources" are (p.9)
This [Gathering] Action cannot be taken directly on the Island tile with the players’ Camp since players get resources from that tile during the Production phase.
You can gather only once per round from each source; i.e. if one player assigns his pawn(s) to gather from one source, no other player can gather from that same source this round.
Addtionally, from the FAQ 1.0 (made by Ignacy Trzewiczek himself) (p.10)
If I put one of my pawns on any Action, will the second pawn placed there automatically support me? Or could we, for example, place 3 single pawns on the Weapons track, to build 3 Weapons (and roll the dice three times)?
It's your choice if you want to treat each pawn separately.