The spell's school is whatever the duplicated spell's school normally is

The basic use of this spell is to duplicate any other spell...

Duplicate here mean to "make or be an exact copy of". Note that the wording says the entire spell is duplicated, not just its effects. An exact copy of a spell means that it is identical in every aspect, which includes school.

If wish was intended to only take some of the characteristics of the spell and not others it would say so. In fact, it does say so, but only for two specific things.

Wish specifically indicates things that are not duplicated from the spell

You don't need to meet any requirements in that spell, including costly components.

Wish tells us that only one thing is different casting a spell using wish versus casting it normally: requirements of the spell do not need to be met.

...The spell simply takes effect.

Also, it is implied by the wording "the spell simply takes effect" that wish also overrides the casting time of the duplicated spell.

Since no other exceptions are noted the spell should be treated exactly as if the spell had been cast normally including duration, range, school, etc.

Wish only lasts for an instant

Wish is an instantaneous spell which means no matter what it is gone the instant after you cast it. But say you used wish to cast a spell, say geas, on a creature using wish. The instant after wish is cast the only effects remaining are going to be of the spell it is duplicating not wish itself.

Wish conjures a duplicate of another spell

Essentially, it seems that using wish to duplicate another spell would result in the casting of a conjuration spell with an instantaneous duration which then results in another spell taking effect with all the details that that spell has (including school). Wish conjures the other spell into existence and then disappears leaving only the conjured spell's effects.

This doesn't work (Option 3*).

The provisions you mention for Teleportation Circle and Guards and Wards are part of the spell effect, not a requirement to cast the spell. You aren't casting a different spell or even a different mode of spell when you cast it that final, 365th time. Rather, you cast the spell as normal, and then because of when and where you cast it (in a place where that same spell had been cast 364 times before), something extra happens on top of the normal spell effect.

You could almost certainly create a permanent portal or warded area using the open-ended version of Wish, though of course that would entail the 33% chance of not being able to cast it again as usual.

As for casting the spell through Wish, that wouldn't work by RAW. By the wording on Wish, you aren't actually casting Teleportation Circle or Guards and Wards, you are casting Wish, with the effects of the other two spells swapped in. "this spell" will refer to Wish when you duplicate it this way.

*However this does open up an extra method. If you cast Wish in a location every day for 364 days, then you can use it to cast Teleportation Circle and/or Guards and Wards on the 365th (or later) day to get the permanent effects of those spells. Notably, it doesn't matter what spell you use Wish to cast, as long as you cast some spell each day. You could even use the non-duplicating version of Wish if you so choose, as long as you don't lose the ability to cast it of course.

So for example, you could spend 364 days using Wish to cast Wall of Stone on the same spot, creating a wizard fortress, and then on the 365th and 366th day use Wish to cast Teleportation Circle and Guards and Wards to permanently create those effects in your new home. You could even keep casting these on later days to set up additional Guards and Wards and/or Teleportation Circles if you wish, as long as you cast them from the same position.