If the function goes horizontal sometimes, then it is nondecreasing. If always, then it's constant (at least if continuous).
Much jargon works this narrowing way, to demarcate that the concept should not carry the everyday cluster of associations and meanings, but rather a domain-specific meaning.
The monotonically is important because it says that at every zoom level the function is increasing, while it being increasing just says it about the function shape in general.
Any function that decreases over any interval of its domain is not defined as an increasing function, rightly so.
A function that gets large with increasing independent variable can be called an asymptotically increasing function under the right conditions.
Not at all. A function can be increasing over a given interval without that increase being monotonous.