There is a precise word for the "actively drinking" action, but it's fairly dated / poetical today. From the full Oxford English Dictionary (behind a paywall, I'm afraid)...
(Section 5 - Senses relating to the imbibing of liquid, subdefinition 14a)
The drawing of liquid into the mouth or down the throat;
an act of drinking, a drink;
the quantity of drink swallowed at one ‘pull’.
From the same source, there's also...
swig (slang or colloquial)
An act of ‘swigging’;
a deep or copious draught of a beverage, esp. of intoxicating liquor; a ‘pull’.
So - if your context is quite formal, draught is exactly what you want. But in any other context you should probably go with pull even though it's a bit slangy. Note that even the OED had no choice but to use it in their definitions - but at least they acknowledge its "lowly status" by putting it in "scare quotes".
Note that I can't see any possible use for a word to reference "the pause between any two consecutive draughts", just as I doubt any language (not just English) has a dedicated word for the pause between any two consecutive acts of putting food in one's mouth.