The question asks for a detailed analysis of Taiji as it compares and contrasts with other martial arts. Looking beyond the specifics, there's an underlying question here about how Taiji is different from other martial arts. If we understand that, maybe it will help answer the specific questions.
The first thing to realize about Taiji is that it's an "internal" martial art. What does that mean? It refers to something called, "internal mechanics". No, it's not some mystical "chi" kind of thing, although a lot of people make it out to be that. Instead, internal mechanics refers to a way of moving the body that's different from normal, external means. Internal mechanics is mechanical and physical. It has nothing to do with chi and some mysterious energy.
In Taiji, there are the "jing". The jing can be defined as "qualities of motion". There are many jing in Taiji, as many as 20 of them from what I gather. Some say the count is 36. I don't think most people learn more than a dozen of them, if they're lucky.
The most famous 4 jing are: P'eng, Lu, Ji, and An. Those are the 4 primary jing practiced during silk reeling exercises from Chen style Taiji. They're also referred to as the 4 "main directional" jing.
Then there 4 other jing that are called the "corner directional" jing which combine with those to form 8 primary jing: cai, lie, zhou, kao.
Don't get too caught up in why they're called the "main directional" vs. the "corner direction", etc. Mostly they just mean "qualities of motion". Don't worry too much about how they're grouped and why, at least for now.
Beyond the basic 8 jing, there are more advanced jing. For example, fa-jing, which is the explosive force. You'll also learn coiling jing and bounce jing. And many others. They each have a different quality to their motion.
Why is it called internal, then?
It's "internal" because motion in Taiji is defined by what's going on inside of the body. It's not easy to see from the outside. Specifically, how the bones are aligned, how the tendons and muscles are pulled tight to support the bones, how movement occurs, storing and emitting power, and so on.
A Taiji punch can look the same as a karate punch from the outside, but inside of the body, different things are going on in each case. In external arts such as karate, the parameters of a punch are defined by how it looks from the outside. But in internal martial arts, the parameters of the punch not only depend on how it looks externally but also what is going on internally.
Take for example, p'eng jing. It's the most basic of all the jing. You need to understand that before moving on to any of the other jing. P'eng jing is the foundation for everything else. You can think of it as the "ground force", but in practice it's about aligning your bones from your feet to your knees, to your hips, to your shoulders, to your elbows, and to your palms. If the bones all along that path are pulled tightly and supported by your muscles and tendons, then pushing on the palms of your hands will, to the person doing the pushing, feel like he's pushing against solid rock. That's because the force of the push is channeled down your bones into the ground.
Some call p'eng jing an upward force from the ground. And in Newtonian physics, that's basically what it is. But the upward force is only enough to neutralize the force of the push and no more. Furthermore, it's not you using your muscles to push against the push. It's more like you aligning your bones and making sure they don't move when the push happens. You still use your muscles, not in actively resisting the push itself, but in making sure your bones don't come out of alignment.
P'eng jing doesn't end there. You have to be able to use it while dealing with someone pushing at you from all directions. So how you move dynamically with p'eng jing is part of Taiji training. One clue is how you never want to be extended too much in one direction in Taiji. If you look at a joint like the knee, you don't want it in a locked position, fully extended. And you don't want it in the opposite extreme, either, where it is fully tucked so that your heels are now touching your butt. Instead, you want it somewhere in the middle. Why? Because then when you are pushed, you have the ability to move to adapt to the incoming forces. If you're extended too much in any direction, you can't adapt, and your structure and balance will be compromised. You must do this with each joint dynamically while maintaining p'eng: knee, hip, shoulder, and elbow.
And that's just one jing. There are 7 more primary jing. And then another dozen or so after that. Each one has very precise qualities of motion that require a high degree of attention and repetition. You can't just do it quickly. You have to do them each slowly, because while you're doing them, you're constantly going down a list of parameters in your mind, correcting yourself as you go. It is hard! And it's why the movement in Taiji must be done slowly at first. Later on, it is sped up as you become more advanced.
So that's why people doing Taiji forms are doing them so slowly. At least ostensibly.
Most people who do Taiji never learn anything about internal mechanics, or they'll learn the wrong things about them. They don't have teachers who know it, basically. Only about 1% or less of people doing Taiji actually understand internal mechanics. And it's because it's mostly practiced as a health exercise. The martial aspects have been lost in a lot of lineages. So you have to find a teacher who knows it and can teach it. I learned more in 30 minutes from one of my Taiji instructors than in the 3 years I trained with other instructors.
The forms are there for you to practice internal mechanics while doing something martial. That's it. To some, the forms are a library of fighting and self-defense techniques. They are, but if that's all you're exposed to, you're pretty limited in terms of your techniques.
So at some point in Taiji training, you start to learn push-hands. Push-hands is an exercise, not the same as sparring. It's just to let you work on some of the jing with a live opponent. At first you're standing still. Later on, you'll move around. And there are even push-hands competitions where it almost looks like sparring.
Sparring is the final thing Taiji trains. In sparring, you're exposed to punches, kicks, blocks, and throws.
While you're practicing Taiji, you're supposed to move in a Taiji way. In other words, a way that still conforms to the parameters of the different jing. It's very difficult. It requires a good instructor and a student willing to work hard for years, maybe even decades.
Now to answer some specific questions. Taiji is neither hard nor soft. It can be one or the other. It just depends on how you're using it. The principle of yielding to strength, redirecting it, following it, and overcoming it is inherent in push-hands practice. But then, when you can feel your opponent's center, you can emit explosive force (fa-jing) suddenly to cause your opponent to go flying. That's not soft! Although, some might say that the sensing aspect of it is what makes it soft, and then the emitting is what makes it hard.
Closed fist, opened fist... It doesn't matter. Taiji does both. What matters is how you're using them.
There are high kicks in Taiji. But pay attention to the body's structure when they do that. They're not doing contemporary wushu style kicks. They're having to rock their hips forward and bend at the knees somewhat. They don't want their body vulnerable to pushing during a kick. They have to be able to adapt and recover when things go wrong.
The focus of Taiji is not necessarily on grappling, but that is a big part of it. They still do free sparring, so they know how to punch, kick, and block. But Taiji is most comfortable in close, making connection with their opponent. Once they're connected, they can sense (through touch alone) what their opponent is doing.
One of the primary goals of grappling in general is to unbalance your opponent and make him fall to the ground. They can die from impacting their heads on the ground. If they don't fall, it takes time for them to recover their balance, and meanwhile you can hit them while they're distracted.
There are plenty of similarities between Taiji and grappling styles like wrestling, Judo, and Aikido. Taiji practices throws, standing joint locks, ankle and wrist locks, chokes, etc. But they'll do them in a way that conforms to the parameters of the jing.
Many Taiji instructors simply don't know much grappling, because to learn it, they really have to get with other people that know it. And typically in Taiji, you need to spend at least 10 years doing the basics before you branch out and start learning all about wrestling, throws, and so on. It's very common for Taiji instructors to also study other martial arts and then take that back into their Taiji training.
I'm not sure if I answered all your questions. I think these questions are pretty common ones that people ask whenever approaching a new martial art. They want to compare what one style does with another style. In the case of Taiji, you really can't compare too much without first trying to understand internal mechanics.
I'll just end this by saying that Taiji is very principled. It requires rigid adherence to the parameters of the jing at all times. It takes decades to reach a point where you have actual fighting skill. If you have an excellent instructor, you can make rapid progress and actually reach a point where you have some fighting ability in just 10 years. Some never reach that point. So if you're interested in Taiji, keep that in mind. There are quicker ways of getting good at fighting. Each style has its positives and negatives.
Hope that helps.