Is a small amount of vomit excused, in terms of its impurity, by analogy with a small amount of blood? And is the relied-upon opinion regarding vomit that has not changed, that it is pure or impure? And does water become impure by the falling of a very small amount of blood into it?

A small amount of vomit is excused. This is a narration from Imam Ahmad, but it is not the relied-upon position in the madhhab. However, there is no harm in acting upon it, especially when there is difficulty. As for vomit that has not changed [its characteristics], it is pure according to the Malikis, and this is a strong opinion. A small amount of blood, if it falls into water, contaminates it if we say that a small amount of water becomes impure upon contact with impurity, and a small amount of blood that falls into it is not excused.