Crimson Hellkite vs Telekinetic Sliver + Darkheart Sliver
If player A taps his Telekinetic sliver to tap opponent's Crimson Hellkite, the opponent can pay 2 red mana (he can pay X, but let say he pays two) and tap the Crimson Hellkite to put on the stack "Crimson Hellkite does two damages to target creature".


Correct.


  Now what if player has a Darkheart sliver in play and wants to sacrifice his Telekinetic sliver to gain three life? Can he do it? Does this goes on the stack too and resolve first?


Yes.


  And then the opponent's ability on the stack fails because there's no target anymore?


Yes.


  And then player A cannot tap the Crimson Hellkite because it is already tapped?


The ability played by Player A will still resolve, it just won't change anything, since as you say, the Hellkite is already tapped.


  "Each player chooses the order in which his or her triggered abilities are put on the stack"


There are no triggered abilities in the scenario you outlined. They were all activated abilities. This line does not apply. See 603.1 in the comprehensive rules: 


  603.1. Triggered abilities have a trigger condition and an effect. They are written as "[Trigger condition], [effect]," and begin with the word "when," "whenever," or "at." They can also be expressed as "[When/Whenever/At] [trigger event], [effect]."