The traits of the plane you're on affect you; the traits of the plane you left don't matter on that new plane
Any extradimensional space is also its own demiplane. The Dungeon Master's Guide on Demiplanes says
This catch-all category covers all extradimensional spaces that function like planes but have measurable size and limited access. Other kinds of planes are theoretically infinite in size, but a demiplane might be only a few hundred feet across. Access to demiplanes may be limited to particular locations (such as a fixed gateway) or particular situations (such as a time of year or a weather condition). Some demiplanes are created by powerful magic, some naturally evolve, and some appear according to the will of the deities.
In the D&D cosmology, also known as the Great Wheel, the
planes are connected in a specific fashion, as depicted in the diagram on page 153. (The diagram does not show demiplanes, because the location and even the existence of these extradimensional spaces is constantly changing.) (147)
That means when a creature travels from wherever it is to an extradimensional space, the creature is literally traveling to another plane. The plane the creature was on? The creature's not on that plane anymore. And the plane the creature's now on? The creature is only there. It's very rare when the plane that was traveled from affects the plane that's traveled to unless the creature somehow brings that effect with it or whatever.
A whatever: portal seepage
Underdark on Portal Seepage says
A newly created portal functions well and sustains a solid barrier between its origin and destination points. As centuries or millennia pass, however, a portal can decay or malfunction…. In addition to malfunctions, portal seepage may occur in older portals. When this phenomenon occurs, qualities of the portal’s destination side start to soak into its origin side.
When a portal seeps, the planar traits described in Chapter 5 in the Dungeon Master’s Guide begin to affect the surrounding area. The rate can vary, but the area covered by the seepage averages a 5-foot radius around the portal per 100 years of age. (53)
However, because creating a portal is so onerous, the idea of creating a portal on a plane possessing a fast time trait then awaiting the portal's decay so that the portal's destination side can benefit from the fast time trait is an exploit that's exclusive to high-level characters. Further, it's unreliable, and it carries with it a significant risk of getting hit by hurled dice.
So when a wizard casts rope trick and climbs up the rope, casts Mordenkainen's magnificent mansion and steps through the door, or—while wearing a necklace of adaptation—climbs into his well-appointed portable hole for a nap, that wizard's traveled to a different plane, and the planar traits of the plane the wizard departed don't have any affect on the (maybe very small demi)plane the wizard's on now, which has its own separate set of planar traits.
- It's still sometimes possible for a joker to mess with the wizard's method of transportation to the new plane from the wizard's origin plane by, for example, by dispelling the trick, by dispelling the mansion, or, because the owner for some reason left it sitting around, by dispelling the hole (rendering it inert for a 1d4 rounds) or by simply lighting the hole on fire.
- Unless a plane says it's otherwise not normal (like the limited air supply present in the extradimensional space of a bag of holding, for instance), this DM assumes the plane's traits are normal. Thus, for example, the demiplane created by the spell Mordenkainen's magnificent mansion has normal gravity, time, and magic traits (DMG 147, 148, and 149, respectively)… which are, as described above, unaffected by the planar traits of the plane on which the mansion's entrance is created.
- Also, because it's important, this DM tries to limit ambiguity by applying the rules for portals to extradimensional spaces: A creature's either on a plane or on a different plane and never on both planes simultaneously. (And, yes, the spell blink sort of says Hi! and I don't care.) For example, stuff comes to the top of a bag of holding as the holder desires so reaching in isn't a thing, and upending a bag spills its contents, and a creature climbs into a portable hole to fetch stuff from it. In my campaigns, this prevents, for example, one PC from carrying another PC around in bag of holding with just the other PC's head sticking out of the bag while the buddy explores a fast-time trait plane just so the other PC can see if his head and his body age at different rates.