The offer from the Lamanite king is remarkable. His own people abandon their lands to allow Zeniff and his people to come in and settle. The fact that people were displaced, as well as the information in verse 8 that they began to build buildings and repair walls, suggest that they were given an area that had perhaps fallen into disuse. The idea that the Nephites from Zarahemla were given their old homeland of the city of Nephi cannot be supported by the textual evidence.
The city is called Lehi-Nephi, which may or may not have been the name at the time the people of Zeniff arrived. It is certainly more of a Nephite name, and so specific to Nephite interests that it would have been unlikely to have been preserved under a Lamanite regime.
After the people of Zeniff arrive, they plant and “did begin to multiply and prosper in the land.” We saw this type of comment in 1 Nephi 18:24 and 25, where Nephi similarly marks the new prosperity by noting that they had the means to live. That is similar to what happens in this verse. The new people were able to grow the crops needed in order to live.
The fact of the planting is more important than finding references to wheat and barley, or to determining the meaning of neas and sheum. There is not accepted correlation between neas, sheum, and known grains. We may only speculate that they are consumable grains based on the rest of the list in which they are included.