Basically you're right, but these specific cases lopen te [something] and zitten te [something] are used quite loosely/freely. A lot more than staan te [something].
Because of this loose usage, it's also used sometimes for more abstract things like: hij loopt/zit te zeuren = he is whining (no need for walking or sitting to be involved), but hij staat te zeuren = he is whining while standing. And in any case the simpler form hij zeurt/hij is aan het zeuren is also correct for he is whining.
If all this is too vague for you, you can definitely keep things simple and stick to translating hij loopt te kijken as he is watching while walking. But don't be surprised not everyone is doing it exactly like that. :)
Since 'lopen' is used to form the progressive present, and you can 'go' by walking, does 'lopen' connote a sort of 'to continue to do' or 'to go on doing something'?
For example: 'Ik zing': I'm singing. Maybe I just started now, and I stopped to answer when you asked what that horrible noise was, but: But would 'Ik loop te zingen:' suggest more 'I go on singing; I have been singing and I will continue to sing'?
Or am I miles off here?
From what I've seen/asked/etc., since all my Dutch comes from DuoLingo...
Zitten/liggen/staan te are more informal spoken Dutch than official book Dutch, and you can just say Hij doet iets or Hij is iets aan het doen in every case and it would be correct. But when we talk, we bend the "rules" a bit, to add a little flavor and offer shades of meaning, so even if you don't say hij zit iets te doen, you'll still hear it.
In English we typically lump zitten/liggen/staan te all under the present progressive: "He is waiting." However, we could form similar sentences in English; they might sound a bit fancy or poetic, but they're still grammatical: The dog sits waiting at the door. He stands waiting impatiently for the bus. She lay waiting for the sound of footsteps to herald his return.
Staan te and liggen te will specifically be used to convey whether they are standing or laying while doing something. Zitten is a bit more flexible, but I don't think you'd say ik zit te lopen or ik zit te staan.
Lopen seems to suggest that it's an ongoing thing, roughly like "keeping on," "going on" or "carrying on" in casual English. Again, we can substitute "is" in those cases: "She keeps/carries/goes on walking" and "she is walking" both convey the same fact, but "carry on" gives it a different flavor and emphasizes that it's an ongoing activity.
In case of travelling verbs (lopen, vliegen, rennen, varen, rijden, etc.), just stick with the aan het doen construction to avoid confusion.
They are not equivalent. In this case you can picture a man walking around a store looking at what they sell. The walking is done in order to look around, the emphasis is on "kijken". If you would want to emphasize "lopen" you would say "Hij loopt om te kijken". These auxiliary verbs are used literally with concrete verbs like "kijken".
In the example Susande gave about 'whining' the construction that "lopen te" or "zitten te" means 'continuing to' you are more free to choose.
I am afraid this is entirely in the imagination of the people at www.duolingo.com I dare everyone to type this in Google Web Search ""Hij loopt te kijken" https://www.google.fr/search?q="Hij+loopt+te+kijken" or in DuckDuckgo https://duckduckgo.com/?q=%22hij+loopt+te+kijken%22 And if you still are not convinced, try the Dutch google web search https://www.google.nl/#q=%22hij+loopt+te+kijken%22
verb + te + another verb
This construction occurs with the verbs (zitten. lopen, liggen, staan and hangen) (to sit, to walk, to lie, to stand, and to hang).
The meaning can be that it indicates the position in which something occurs (and it is always ongoing).
De appels hangen te rotten aan de boom. The apples are rotting on the tree. (while hanging)
Hij staat de vaat te wassen. He is washing the dishes. (while standing)
Zij zitten tv te kijken. They are watching television. (while sitting)
However, the meaning can be weaker and the meaning of the position in which something is done is lost. As a native Dutch speaker never thought of this before, consciously, until my girlfriend had these questions on duolingo. I myself have the idea that the differences between zitten, lopen, liggen and staan (hanging does not have this weak form), is maybe a bit in the nuance. In order lopen staan zitten liggen there is slightly more activeness.
Ik loop me te vervelen I am being bored.
Ik zit me te vervelen I am being bored.
The first, with 'lopen', may express a little bit more annoyance. Possibly the person is bored by their own action, their incapability to do something, they are continuously wandering around in their brain to to find something fun to do. The second meaning may be expressed a bit more in the situation in which a person is in a passive situation. For instance watching an event and not finding it interresting. So I'd say:
I am being bored (I have nothing fun to do and are working on getting an idea) - Ik loop me te vervelen
I am being bored (My current activity is not interresting) - Ik zit me te vervelen
I am being bored (I don't have energy) - Ik lig me te vervelen
Quite often those situations with being bored coincide with the positions as well. E.g. being bored while lying on the couch or being in bed. It can be weaker. Combinations with, vertellen (to tell), roepen (to shout), beweren (to argue), zingen (to sing), kijken (watching), zeuren (to whine).
Zeuren is another good case in which the nuance may be strong. For me the meaning of 'hij staat te zeuren', 'hij zit te zeuren' en 'hij loopt te zeuren' is a slightly different. The first, with 'staan', expresses more someone that blocks the way and could be expressed if someone is anoyed about that (when someone keeps complaining and blocks a continuation, 'Sta niet zo te zeuren!'), the second with 'zitten' expresses someone that complains in a situation while not doing much about it or not being much involved (e.g. an observer of a soccer game complaining about the players), the third when someone repeatedly complains coming back on some issue many times ('hij loopt al de gedurende de hele cursus te zeuren dat de Nederlandse grammatica lastig is' )
http://www.ict.teno.be/haasrode/neandertaal/NedLer09.html (this link includes the verbs: proberen, beginnen, vergeten, durven, en hoeven and they also occur in English 'You don't have to try to forget to dare to start to speak Dutch')
For understanding the change of the meaning of position or physical state. The English language has the same. For instance 'the computer is running well'.
Thank you for the detailed explanation. Just want to point out that "Ik loop/zit me the vervelen" would be translated as "I am bored" in English, not "I am being bored". When you are "being" bored it means someone or something is boring you. Even then, you wouldn't use "being" - e.g. "I am bored by this movie".
It depends on the verb that comes with the sentence.
You wouldn't say 'Ik loop te zwemmen' (I am swimming), but 'Ik ben aan het zwemmen' (I am swimming) is right.
With a verb like 'fluiten' (to whistle) you could say 'Ik loop te fluiten', 'Ik zit te fluiten' (I am whistling) or 'Ik ben aan het fluiten' (I am whistling).
Also with reflective verbs like 'zich vervelen' (to be bored) you could use all three forms: 'I ben me aan het vervelen', 'Ik zit me te vervelen' and 'Ik loop me te vervelen' ( I am bored).
They are however all less charming ways to express the action and used more in spoken than in written form. It is nicer to just say 'Ik zwem', 'Ik fluit' and 'Ik verveel me' but they would literally translate to 'I swim, I whistle, I am bored'.
I reckon which form to use has no grammatical rules, but needs getting used to.
Not a native, but my understanding is that there's three ways to express the present continuous in Dutch:
Using the simple present and relying on context. If someone asks "wat doe je nu?" (which is also in the simple present tense), it's implied that your answer is an explanation of what you are doing right now. So if you reply with "Ik eet," nobody is going to get confused as to whether it's a habitual action, or a hypothetical answer, or a statement of what is currently happening. This is pretty common, and you can go through life using only the simple present and sound fine. But in order to understand other speakers, you still have to learn the other two.
[zijn] aan het doen. This emphasizes that the action is currently being done, perhaps as an excuse or explanation. If you want to explain why your child can't come to the phone right now, you could say "ze is haar huiswerk aan het maken" (she's doing homework) or why you are running late--Sorry, ik ben op mijn man aan het wachten (sorry, I'm waiting for my husband). It's a bit like adding "...in the middle of..." to an English sentence, to underscore the fact that the action is currently ongoing.
[zitten/liggen/staan/lopen] te doen. This is apparently far more informal than the other two. It just means that someone is doing something in a given posture. I guess you could think of it in English as when someone says "he just sits there and plays games on his cellphone" or "don't you stand here and accuse me of cheating!" The main point of those sentences isn't that they're sitting or standing, but that they're playing games or accusing you. But adding the posture helps "paint a picture" a little bit.