Eating/Drinking/Arena Map

IC, the characters don't know the actual layout beyond what they can see. Eating / Drinking / Etc is not fully based off a dice roll; there is a Tribute Maintenance thread where tributes' actions for the day are posted (e.g. 'got matches from sponsorship, lit a fire, boiled both jugs of water, drank one'). Eating / Drinking are only rolled for when there is some doubt as to the purity of the items (e.g. water that has not been boiled/purified, plants that have not been identified as safe.) Rolling for these are done in threads.

The tributes aren't, but the members who play them are — tributes would still be able to decide to go north/east/south/west/etc, so just because they don't know what they're heading toward (necessarily; some landmarks are visible from other areas) doesn't mean they can't go there.

The people who play the tributes choose what sponsor gifts they receive with whatever sponsor money they have.

That is correct, yes. Some areas do go unexplored.