"Room" versus "space" (in the sense of unoccupied volume)

Solution 1:

The NOAD defines space and room as follow:

• room: space that can be occupied or where something can be done, especially viewed in terms of whether there is enough
• space: a continuous area or expanse that is free, available, or unoccupied

The difference is that room is used to refer to space that can be occupied, while space is used to refer to an unoccupied area, and which could be kept unoccupied.

Solution 2:

In most of your examples, you could exchange space and room without a change of meaning and without sounding notably unidiomatic:

The general wants room to manoeuvre. There's no room for a new file on my hard disk. There's no space on my shelves for a new book. Maybe there's space for argument here.

("space for argument" is marginal but not totally wrong to my ear.)

The only exception is:

[?] Most people want breathing space.

This is because "breathing room" is a fixed idiom, and you can't freely replace "room" with a synonym without damaging the idiom. There are probably other idioms that don't come to mind at the moment, but outside of specific idiomatic expressions, you can assume that "space" and "room" are interchangeable synonyms when used as above.

However, the more literal uses of these words are clearly not interchangeable, such as when "room" refers to a physical enclosure with four walls, or when "space" refers to the region beyond the Earth's atmosphere or ASCII character 0x20.