Unless stated otherwise, any ability granted to creatures by another creature (e.g. by Gemhide Sliver, Constricting Sliver, etc.) cease to exist when the creature granting those abilities leaves play, correct?
That depends on what kind of effect grants the ability.
If the effect comes from a static ability (such as from Gemhide Sliver and Constricting Sliver), the effect lasts for as long as the object with that ability is in the appropriate zone (usually the battlefield) (C.R. 611.3b, 611.2a, 113.6). For example, Gemhide Sliver and Constricting Sliver each have an ability that works only while that permanent is on the battlefield (C.R. 611.3b, 113.6).
If the effect comes from a resolving spell or ability, the effect lasts for as long as stated on that spell or ability (e.g., "until end of turn"; an example is found in Shaper Guildmage), or for the rest of the game if it doesn't say how long (C.R. 611.2a).
Does the same principle apply to a creature you have taken control of with Nicol Bolas, Planewsalker, that is, does Nicol Bolas, Planeswalker have to be/stay on the battlefield to keep control of the creature or is that a different situation?
Does the same principle apply to a creature you have taken control of with Nicol Bolas, Planewsalker, that is, does Nicol Bolas, Planeswalker have to be/stay on the battlefield to keep control of the creature or is that a different situation?
Unlike control-changing auras like Mind Control and static abilities such as the slivers you mentioned, the control-changing ability of Nicol Bolas, Planeswalker is a continuous effect, making it wholly independent of the actual planeswalker permanent. As a result, if you activated the ability to take control of a permanent, you will retain control of that permanent even if the planeswalker changes or control or leaves the battlefield.
Does the same principle apply to a creature you have taken control of with Nicol Bolas, Planewsalker, that is, does Nicol Bolas, Planeswalker have to be/stay on the battlefield to keep control of the creature or is that a different situation?
When Nicol Bolas's second ability resolves, it creates a control-change effect that lasts for the rest of the game (even after Nicol Bolas leaves the battlefield), since it doesn't say how long it lasts and the effect is from a resolving ability (C.R. 611.2a). The principles I gave in the last two paragraphs in comment 2 are not limited to continuous effects that grant abilities or change control of a permanent.
If the effect comes from a static ability (such as from Gemhide Sliver and Constricting Sliver), the effect lasts for as long as the object with that ability is in the appropriate zone (usually the battlefield) (C.R. 611.3b, 611.2a, 113.6). For example, Gemhide Sliver and Constricting Sliver each have an ability that works only while that permanent is on the battlefield (C.R. 611.3b, 113.6).
If the effect comes from a resolving spell or ability, the effect lasts for as long as stated on that spell or ability (e.g., "until end of turn"; an example is found in Shaper Guildmage), or for the rest of the game if it doesn't say how long (C.R. 611.2a).
Unlike control-changing auras like Mind Control and static abilities such as the slivers you mentioned, the control-changing ability of Nicol Bolas, Planeswalker is a continuous effect, making it wholly independent of the actual planeswalker permanent. As a result, if you activated the ability to take control of a permanent, you will retain control of that permanent even if the planeswalker changes or control or leaves the battlefield.
EDIT (Aug. 15): Edited.