The doll is a kokeshi; I would not know how else it could be called.
The writing on it is clearly the first half (the first four lines below) of the tanka by naturalist poet 若山牧水. (Personally lovin' that extremely long 「リ」 in the second line.)
Why just the first half? You might ask. That is because in Japanese culture, things left unsaid have just as much or even more 'power' than the things that are said. The reader is expected to finish the poem by himself in his heart, appreciating the act of doing so.
I would rather leave the translation to one of the experts here, but here is one by me for starters. If you know anything about Japanese, the original word order will need to be switched around completely for an English translation.
"I have passed so many mountains and rivers,
Wondering if I ever reach a country
Where loneliness ends,
Here I go hiking again today"
If you translate, it is impossible to say exactly what part of it the writing on the kokeshi corresponds to because of the drastic change in the word order. It is just "about the first half".