My answers after playing around a bit.

1. order preserving, \\(f(x) = 2x\\). non order preserving \\(f(x) = \frac{1}{x}\\)

2. metric preserving, \\(f(x) = x + 2\\). non metric preserving \\(f(x) = \frac{1}{x}\\)

3. addition preserving, \\(f(x) = 2x\\). non addition preserving \\(f(x) = \frac{1}{x}\\)

1. order preserving, \\(f(x) = 2x\\). non order preserving \\(f(x) = \frac{1}{x}\\)

2. metric preserving, \\(f(x) = x + 2\\). non metric preserving \\(f(x) = \frac{1}{x}\\)

3. addition preserving, \\(f(x) = 2x\\). non addition preserving \\(f(x) = \frac{1}{x}\\)