Henro Pilgrimage Shikoku at New Year

Hi,

I am a foreigner living in Japan and am planning to do a part of the henro pilgrimage in Shikoku. I want to do around 200 kms in 10 days on foot. The main problem is that I want to do this at the time of new year when it is very cold and Japan is practically closed. So, I have few questions :

1) I want to walk on 29th, 30th, 31st December, 1st January when nearly everything is closed in Japan. So, would it be possible to secure accommodation in a ryoukan or any inns on the henro path on these days?

2) I know it would be around 4 degree C at that time. Has anybody tried to do this in this temperature ? Was it doable ?

3) How does the booking system work for the inns on the path of henro ? Do I have to book all the inns before starting my journey or can i book each 1 day before my stay ?

4) Finding vegetarian food is difficult in japan but since this is a Buddhist pilgrimage, I was thinking that vegetarian food will be available at the inns which come in the path. Am I correct or finding vegetarian food will be difficult on this path too ?

Any kind of help will be deeply appreciated.

Thanks in Advance.