Seasons on a planet that's tidally locked with the smaller star in a dual-star system

Neil's completely right that the question is a bit confusing. There are two scenarios here: The planet orbits in the same plane as the stars, or it orbits 90 degrees perpendicular to them. There's a huge difference between the two setups. I have a feeling, though, that you're curious about the other, and I really can't resist launching into an explanation of them both, so I'll cover them separately.

Situation 1: The same plane.

Is it feasible at all?

Is it feasible? Of course. We know that planets exist in binary star systems, and some may be habitable. At any rate, though, it is important to discuss the difference between planets that only go around one star and planets that go around two. There are two designations for planets in binary star systems, based on their orbits:

Planets that orbit just one star in a binary pair are said to have "S-type" orbits, whereas those that orbit around both stars have "P-type" or "circumbinary" orbits.

In your case, the planet only orbits one star. Nice choice; circumbinary planets (planets that orbit both stars) might have drastic temperature swings, and may orbit far out from the stars. I'd rather live on a planet with an S-type orbit than on one with a P-type orbit.

There's a reason that planets generally form on the same plane as their stars. It's because the system forms from a protoplanetary disk, which flattens out because it rotates. This is characteristic of most star systems. Check out this question for a clear explanation. This question is okay, too, but I don't think the answer properly addresses this issue. There are exceptions to the "same-plane" rule, but at the moment I can't find them.

How cold would the outer hemisphere become in a cold season?

This is hard to figure out because there are a lot of factors that you don't have. For example, do the stars' orbits have low eccentricity or high eccentricity? What are their properties? I'm not sure how to figure this out, given that we only have information for one position.

How many seasons would the inhabitants encounter, and what they would be like?

Currently, you could draw inspiration from this question, which has a lot of interesting answers. Unfortunately, only a few address binary-star-scenarios. Common sense, though, tells me that, if the orbits of the stars have relatively small eccentricities, the planet would have odd seasons. Remember, we have seasons because of the tilt of the Earth's axis. I think the planet would have seasons primarily caused by the tilt of its axis, and that the other star really wouldn't play into it very much. You can find relevant questions on Astronomy here and here. However, the tidal locking could impact the seasons.

Situation 2: A 90 degree tilt.

Uranus springs to mind when I think of this scenario. It's tilted almost 100 degrees from the "horizontal" - or about 80 degrees, from another perspective. Its rings, therefore, can be a good model for this planet. Well, not really. But they can give you a good picture of its orbit.

Is it feasible at all?

I'm going to go with no for this one. Why? Uranus is tilted at such a large angle possibly because of an impact with another object. For this scenario to work, you'd need to have the protoplanetary disk of the primary star tilted 90 degrees, which is implausible.

How cold would the outer hemisphere become in a cold season?

This is rather easy. Assuming the stars orbit each other at low eccentricities, the planet would always be the same distance from the other star, so temperatures would be constant. This means, though, that one hemisphere would be perpetually warm, while the other would be perpetually cold. It would be tidally locked - to both stars!

How many seasons would the inhabitants encounter, and what they would be like?

It follows that the planet would have seasons just like any other tidally-locked planet.

You might find this question interesting. And this pre-print is quite a good read.

Edit:

As per your comment:

I also don't see why the planet should be locked to S. Even so, the outer hemisphere is roughly half of the time exposed to S, and roughly half of the time shaded from it.

Think about it like this:

One pole of Uranus is generally pointed towards the Sun. This is like the axis of revolution of the planet being pointed towards S. Now, imagine the Earth-moon system. From someone looking "down" on the system from "above" Earth's North Pole, how many sides of the Moon are visible? Only one hemisphere. S is like this observer, looking "down" on the other star and planet.

As for capture - sure, the planet could be capture by the system. There are two problems with this:

1. It would be very hard for the planet to be captured by the star and become tidally locked. Tidal locking is generally an indicator that the objects have a pretty close history.
2. Given that S is more massive, it would be more likely that S captures the planet, and not J. Sure, J has a fair chance of getting the planet, but S would probably have bigger influence.

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