The effects move with you.
The 5e SRD Appendix C details how the planes of existence behave. There's nothing to suggest magic behaves any differently in the ethereal plane than the material plane (outside of some spells that have explicit interactions with the ethereal plane). The effect would move into the ethereal plane with you and remain active at your new location. It would cease to function where you used to be.
Project image remains where it was placed because it does not follow you or remain centered on you.
Floating disk cannot move with you as you fly because that case is addressed in the spell description:
It can move across uneven terrain, up or down stairs, slopes and the like, but it can’t cross an elevation change of 10 feet or more. [...]
If you move more than 100 feet from the disk (typically because it can’t move around an obstacle to follow you), the spell ends.
Emphasis mine.
Mirror image and warding wind do not impose restrictions on where the effects can follow you, so there is nothing to indicate planar travel is off-limits.
For most spells, no.
As described in the "Areas of Effect" section of the spellcasting rules:
A spell's description specifies its area of effect, which typically has one of five different shapes: cone, cube, cylinder, line, or sphere. Every area of effect has a point of origin, a location from which the spell's energy erupts.
This point of origin is a point in the space from which the magical energy spreads, but it is not its source.
The description of Antimagic Field says, in part (bold for emphasis mine):
Areas of Magic. The area of another spell or magical effect, such as fireball, can't extend into the sphere. If the sphere overlaps an area of magic, the part of the area that is covered by the sphere is suppressed. For example, the flames created by a wall of fire are suppressed within the sphere, creating a gap in the wall if the overlap is large enough.
Hence, if a Fog Cloud is hovering somewhere and an Antimagic Field is cast nearby intersecting the area of effect of the former spell, then the fog is suppressed only in the intersection of the two areas of effect, even if the point of origin is included in this intersection.
For some spells, yes.
There are anyway some spells for which covering the object which is the origin of the spell suppresses the effects on the entire area of effect. For example, consider the Darkness spell that you mentioned (emphasis mine):
If the point you choose is on an object you are holding or one that isn't being worn or carried, the darkness emanates from the object and moves with it. Completely covering the source of the darkness with an opaque object, such as a bowl or a helm, blocks the darkness.
Hence, if Darkness is cast on an object and the Antimagic Field includes that object, the darkness produced by the spell disappears. Indeed, the description clearly states that the origin of the darkness is the object: blocking the magic coming out of the object via an Antimagic Field blocks the effects of the spell.
This suppression is confirmed by another part of the description of the Antimagic Field spell:
Spells. Any active spell or other magical effect on a creature or an object in the sphere is suppressed while the creature or object is in it.
On the other hand, if Darkness is cast choosing a point in space as origin and the Antimagic Field includes such point in its AoE, then the Darkness is supressed (i.e., there is no heavily obscured area caused by the spell) only in the intersection of the two areas.
The same happens for the Light cantrip:
You touch one object that is no larger than 10 feet in any dimension. [...] Completely covering the object with something opaque blocks the light.
Best Answer
In short, this feature will work on magically created weather
Features in 5e do what they say they do. If the feature was meant to exclude a particular category of weather (like weather created by a magical source) then the spell would describe those exceptions. Since the feature doesn't make any exceptions, if a magical effect creates weather, then this ability can affect it.
With that being said, there is one caveat...
It depends on the DM's definition of "weather"
In 5e, terms that do not have explicit game definitions are meant to be interpreted based on their natural language definitions.
Although a section on weather does exist in the DMG, it doesn't define weather as anything explicit in game terms. So we use the natural definition, which merriam-webster defines as:
At first glance, everything looks fine. But colloquially, not every single atmospheric effect is what I would call "weather." And that's where your sticking point is.
Do these spells actually count as weather?
It seems a silly question, but think about it. If you use a leaf blower to clear your lawn, does that count as weather? If I stir the water in my glass to make a miniature whirlpool, is that weather? If not, what makes those that different from using magic to create these exact same effects only on a larger scale?
In fact, there is even some precedent that not every atmospheric effect is weather. When asked about the use of create or destroy water on a vampire's mist form, Jeremy Crawford, lead rules designer, replied:
A DM is well within their right to rule that one or more spells that have an effect on the atmospheric conditions around a character do not constitute weather. Taking a previous example, it would be akin to them saying "gust of wind doesn't create weather, it's just a magic fan blast. That's not weather." If they make such a ruling, you're out of luck. The feature only works on weather, and your DM said that particular effect isn't weather. However, the DM is just as justified saying "oh, yeah. That's totally weather." and then you're good to go.
TL;DR
Since there's no solid answer on this specific scenario from any official rule clarification (as far as I could find) the answer then boils down to: You might be able to as long as your DM rules that the magical effect really does constitute "weather" instead of magical manipulation of the atmosphere (like a fan on a hot day, only magical instead of mechanical).