Did he give to her just six barley seeds?
Rashi: Yes. You cannot say that he gave to her six Sa'im (the volume of 864 eggs) - it is not normal for a woman to carry so much 1 ! Rather, he gave six seeds, to hint that a son will descend from her who will be blessed with six Brachos - "Ru'ach Chachmah u'Vinah Ru'ach Etzah u'Gevurah Ru'ach Da'as v'Yir'as Hashem" (Yeshayah 11:2).
Malbim: No. Shesh is a Kav, a sixth of a Se'ah. It was common to divide a measure into six parts - "v'Shishisem ha'Eifah" (Yechezkel 45:13). An Oni going from place to place, we do not give to him less than a loaf of half a Kav. He gave to Na'ami and Rus a full Kav; it suffices for the morning meal for both of them. This revealed that he will finish the matter today, and they will not need the night meal 2 .
Why does it say "va'Yavo ha'Ir"?
Torah Temimah citing Rus Rabah 7:3: He accompanied her, lest one of the boys encounter her.