gottoms-
he plays gottoms activates effect tributes whatever i chain enemy controller to his effect now enemy controller resolves first so does my opponent still get the effect??

hyunlei-
same thing chain enemy to his selection of back cards since enemy would resolve first does he still blow up the cards

darksoul-
stacking rule and mutiple in the graveyard how does that work now can i get mutiple searchs??

Yes, Gottom's Emergency Call still resolves with effect since the requirements upon its activation have been met by having an X-Saber monster face up on your field.

Hyunlei follows the same ruling. In order for his effect and Gottom's effect to not work they need to have their effects either negated or the activation of their effects negated.

Here's the ruling for Darksoul for you.

If multiple copies of "XX-Saber Darksoul" are sent to the Graveyard and they are all in the Graveyard during the End Phase, then you can activate each copy's effect.

Source: http://yugioh.wikia.com/wiki/Card_Ru...Saber_Darksoul

For the first 2, unless rulings specifically state otherwise (or the effect is a Continuous Effect), then once the activation requirements are met for a card, then that card does not have to remain face-up on the field for the effect to resolve successfully.

You would have to negate the activation and/or effect of the card to prevent it from resolving successfully.