It is not illegal to change your name, as long as it is not done to defraud. If he wanted to use "Friese Greene", he could (he may have only used it for business initially hence giving just plain "Green" on censuses). If someone preferred to be called Annie rather than Ann, or use their middle name, or take the surname of their stepfather, they could. Most people didn't bother with things like deed polls.
Although some ages changed because people were trying to disguise things like a big gap in age between husband and wife, there are lots of reasons why somebody might be mistaken, particularly for more distant relatives (in-laws, etc) in the household. That's why ages and birthplaces for lodgers and boarders are more often wrong, generally speaking, then those for family members in the census.
A birthplace might also be wrong because someone wrongly assumed they were born in the location where they lived as a child, or because it was easier to give the name of a large town near their actual birthplace.
Following the Josty connection:
http://www.badrutt.org/PiBa-GR-BK2006-e.htmhttp://www.badrutt.org/PiBa-GR-020-e.htmDaniel Josty and his brother were photographers in the Engadine area, but their family also had a connection to Berlin. If the information about Daniel being her step-brother is correct, "Ida Josty" may be Helena's mother.
This may all refer to the same family:
http://en.wikipedia.org/wiki/Caf%C3%A9_Jostyhttp://www.luise-berlin.de/strassen/bez01h/j134.htm (in German)
adressbuch.zlb.de could help find where the family were living in Berlin.