Phew, big question! Let's see if I can address some of that...
"The process" definitely gets harder to talk about as it progresses. This is true not only in my personal experience, but is also substantiated by the tradition. Indeed, if Enlightenment was simple and straightforward to explain, wouldn't everyone be enlightened by now? More seriously, would Buddhism evolve so many different schools with their unique ways to talk about it?
According to most schools, the mind of Buddha is indescribable. Some schools insist Buddha is completely mindless (the consciousness ceases), while others posit that while Buddha does have awareness, it is completely beyond concepts. Buddha himself said:
A 'position,' Vaccha, is something that a Tathagata has done away with.
If the above does not explain, it at least provides some justification for why the talk about Enlightenment can hardly not be metaphorical, beating around the bush, and involving multiple mutually contradicting descriptions.
Is it gradual?
Yes and No. It is gradual in the sense that it involves work and unfolds increasingly over time... It's not "gradual" in the sense that it can not be achieved by doing the same thing again and again mechanically, and waiting for it to just happen. (Google "Zen polishing the tile" story.)
One of my teachers said: "Enlightenment is a choice we keep on making". This highlights one aspect of it, that it's not necessarily asymptotic nor is it irreversible, but because it's "a choice" we must learn to "keep on making", there is a sense in which Enlightenment is a skill that is developed over time.
You do know that this very question (gradual vs. sudden) was the main point of debate between the schools of Chan, as well as between Chan and Tibetan Buddhism? Although, some modern scholars conclude that such a reified distinction between schools was more like a rhetorical device used by the subsequent schools, and that none of the historical schools (to an extent that we can even talk about distinct "schools") were in fact fully "sudden" or "gradual".
Is it more and more moments?... there's no such thing as an "enlightened person"...
I would rather characterize it as "less and less" rather than "more and more". In my experience, it is less and less grasping/attachment, less and less overgeneralization, less and less confusing abstractions with reality, less and less losing emotional balance. And yes, less and less "enlightened person", less and less thinking/worrying about oneself in general, less and less trying to characterize oneself as something specific (enlightened or whatever). Less and less thinking of oneself as one distinct entity or agent. Less and less conflict, less and less suffering. It's like, the tangle gets progressively looser, until it's barely in a single piece anymore. Or, a fire that is pulled apart, into individual pieces of wood...
I was wondering what the difference is between [sotapanna] and the "higher" stages of enlightenment -- and how to achieve or progress towards those, what (what practice and/or realization) is required?
I think it requires the kind of practice that Buddha metaphorically described as leveling of the earth, starting from the mountains, then hills, then progressively smaller features, until even the ground itself is "flattened away", followed by space and the very mind itself. The way I understand this, we should keep looking for obstacles to suchness - the sources of existential conflict that generate the experience of suffering.
The sense of "I" is just one of such sources (albeit a major one). Another big source of suffering, recognized in Mahayana, is the idea of Enlightenment itself (aka the idea of Nirvana as goal "over there"). At some point the practitioner is supposed to "get over" (to quote my Zen Master) the notion of the goal, because the very notion of the goal dependently co-creates anxiety of not having reached the goal. Just another "bump" to be "leveled". Then there is this idea of reincarnation or rebirth, which is really just the good old Self in disguise. And then there are myriads other "bumps" not necessarily as universal but equally important for an individual practitioner. Someone is hung up on the unrequited love, someone has a problem of being self-conscious and shy in the social setting, someone is passive-aggressive etc. Every one of these is supposedly rooted in some sort of attachment/overgeneralization that must be transcended on the way to complete Enlightenment.
In this light, achieving progress towards the higher stages definitely requires going beyond the basic morals/meditation/philosophy and into the nitty-gritty of our everyday life situations and hang-ups. Like, definitely, 100%.
Does the "dhamma-eye" being associated with stream entry imply that "the Dhamma" is already seen at that stage...
As I said in other answers, I think that stream-entry is by definition the point when one gets more or less clear about the problem, the goal, and the method - and hence "enters the stream" of right practice solving the right problem, leading to the right goal. With stream-entry, suffering has been directly recognized as subjective condition, relationship between craving/attachment/overgeneralization and suffering has been directly seen, the cessation of suffering has been directly realized in at least one real case, and -- most importantly -- the overall way "things work" has been intuitively grasped.
Practice subsequent to stream-entry may certainly involve new discoveries and realizations, it's not like you stop learning - but all these discoveries and realizations should all fit within / feed back into, this basic framework of 4NT and "the way things work".
In a recent answer I said, stream-entry is when a student is supposed to completely get it 100% with no doubts, unlike Buddha who has to figure it out from scratch having doubts until the last moment. But in practice that 100% point-in-time understanding was only possible back when Buddha was still alive and Dharma was delivered straight from the source. As time passes between the lifetimes of Buddha and the student, the Dharma gets gradually more "corrupted" (diluted, mixed-up), and the phenomenon of stream-entry gets gradually less straightforward. The future students of Buddhism must do more and more work by themselves to figure out the puzzle, which makes their paths increasingly closer to that of the Buddha, at least in this one sense.
, and thence can only become more ever-present (more of the same)? ...is that related to so-called "mindfulness"?
In my experience, as Dharma is internalized, and the habit of grasping is reduced, what I call "hard mindfulness" (keeping a constant watch over oneself) gives way to something much more fluid, more like the effortless ice skating. So, if we can speak about "more of the same" - it's more of the same fluidity and effortlessness. It's not that due to mindfulness your "moments of enlightenment" keep growing until it's one continuous Enlightenment. It's more like, due to the letting go and non-grasping, the GAPS between samsaric moments occur more and more frequently, until the whole thing gets so loose that it's not there anymore. Chogyam Trungpa Rinpoche spoke about this at length, the porous nature of reality, the soda-bubbles, the ping-pong, the gaps, and the space dancing with itself. I think it's a nice way to describe it. In fact, one of the biggest virtues of Tantra (in my opinion) is providing a language to talk about these indescribable experiences in a way that can be shared and understood.
Why does Right Concentration (samma samadhi) talk exclusively about jhanas? Are they, how and/or why are they, at what point or stage are they, relevant and necessary for any (further) progress?
My teacher said that in the Eightfold Path, concentration (Samadhi) is not your regular "concentration" as in focusing on an object. It's more like ability to shift your attention of your subjective world to achieve a particular perspective on things. For example, we can look at things from a biological standpoint, or a sociological standpoint, or a semiotic standpoint, or a Buddhist standpoint. According to my teacher, in the Eightfold Path, concentration (Samadhi) requires developing an ability to control our perspective/attention of how we see the moment and life at large.
Specifically in Buddhism, first we develop ability to see the world phenomenologically as The Six Senses, Five Skandhas etc. Then we step it up a notch and develop ability to see Emptiness of all phenomena and the subject. Finally, we develop ability to let go of any position and sustain nirvikalpa-jnana, which is a completely unreified / unovergeneralized mode of awareness. Apparently this is what's actually referred to as Samadhi in the Eightfold Path.
Can you outline any connection between practising jhanas and living/acting/thinking in the world?
This is a big topic, but basically, the practice of jhana manifest in the everyday life as positive, upbeat attitude. You are supposed to understand the mechanisms of mind enough to be able to always maintain yourself in a good mood.
So there's all that on one hand... Now, let's put all this aside and talk about the following. In one sense, everything is already perfect. This includes the world and all sentiment beings including yourself (myself) in whatever state they are. Everything is perfect just as it is, with all its imperfections, that's a fact. Achieving a direct clear vision of this is known as the Sudden Enlightenment. It has an effect of completely liberating one from all suffering. After all, what suffering can there be, if everything is perfect? So when the Sudden Enlightenment schools make fun of gradualists, they understandably contrast the never ending perfectionism of gradualists with their own nice and simple way to be free, if only one can see it directly. Everything is perfect. Enlightenment is a choice we keep on making.
There's an opinion that Anuttara-Samyak-Saṁbodhi of the Buddha entails having both views of Enlightenment, gradual and sudden, coexist in one mind at the same time, with no cognitive dissonance whatsoever. I guess this is why Buddha's mind is said to be indescribable?