Is there a general word for inhaling and exhaling ? Something like ''haling'' or ''halation''?

Solution 1:

No.

halation exists

OED

n. 1. Photography. A halo-like effect in which light spreads beyond the edges of a bright object in a photograph.

Haling exists:

OED

hale, v.1

Now chiefly superseded by haul v.

1.a. transitive. To draw or pull (a person or thing) along, or from one place to another, esp. with force or violence; to drag, tug. Frequently with adverb or prepositional complement.

The word is

OED

respiration, n. 1.a. The action of taking air into the lungs (or water into the gills) and expelling it again, esp. as a continuous physiological process; breathing;

As far as I can see there is no English verb that uses the Latin root verb hālāre to indicate "breathing". There was one in Middle English "anhele" but that related to puffing and panting.