Omitting "by" preposition and the resulted phrase

Welcome to ELU and thank you for a fascinating question. I am not sure that there exists a decisive answer to it.

Your examples 1 is the best example of what you are asking about. You are right to say that if you take away the 'by' you seem to make no change to the meaning of the sentence, even though you do make a change to the supposed grammar of "using' from being a noun (a so-called gerund)governed by the preposition "by") to being an adjective (a so-called participle) agreeing with the subject. 2 is more doubtful. The removal of the preposition does not, in my view, alter the meaning in the way you think it does. It does make the sentence sound odd, even so. Your third does seem to detach the word dance from "watching" to "I learned", or rather, again, it renders the sentence grammatically ambiguous: it could be how I learned or how I danced, though common sense and experience make the meaning intention obvious.

Part of the problem is that speakers/writers of English have come to use a single form (a base verb followed by the termination '-ing) for what, in the ancient language from which much of our formal grammar is descended (Latin) is served by two separate forms: the participle and the gerund. Unlike the English '-ing' word, however, neither the participle nor the gerund ever forms part of a prepositional phrase, and the grammatical roles of both participles and gerunds are determined by case. By case, as I am sure you know, I mean the different endings used to define whether they indicate 'agent' (nominative), object (accusative), possession (genitive), indirect object/interest (dative), cause/origin (ablative). For a long time the language of literacy in England (Latin) was different from the language of everyday speech (anglo-saxon English).

Let me illustrate with an example slightly different from your sentence no. 2. (to make the Latin easier.

2a I informed my friends by sending letters.

This could be expressed by the use of a participle in Latin as:

2ai Amicos meos epistulis missis certiores feci.

The participle missis is the ablative of the passive past participle of 'mitto' (I send) and agrees (in gender - feminine, number - plural, and case - ablative, in this instance, ablative of cause or method) with epistulis. So literally it means ' letters sent'. But the simplest and most direct possible translation into English is 2a. If you remove the preposition 'by' from 2a, it would make no difference to the Latin, which neither has not needs a preposition. The case carries that meaning.

Sentence 2a can also be expressed by means of the gerund 'mittendum', a noun meaning 'a/the sending'. This is a neuter, singular only active noun. So the Latin would be:-

2aii Amicos meos certiores epistulas mittendo feci.

This means exactly the same as 2ai. But this time the noun for 'letters' is accusative, because it is the object of the gerund 'mittendo'.

But there is even a third way to do this, less often spoken of. Latin has not only a gerund but also an adjectival equivalent, called the gerundive. It works fine in Latin, but can only be translated into English by very forced wording. So 'mittendus' (feminine 'mittenda', neuter 'mittendum') means something like 'to be sent' or 'fit/ought to be sent'. The most famous use of this is the brutally blunt judgement brought back to Rome from North Africa in 152BCE by Cato the Censor:

Delenda est Carthago - Carthage should be destroyed.

In the more peaceful context of 2a we should get

2aiii Amicos meos epistulis mittendis certiores feci.

This is a more far-fetched but nevertheless possible version of the same English, and means, in its context, much the same: by the sending of letters or by letters being sent

English has only the verb with a final '-ing' for all this apparatus of alternatives. Sometimes, for that reason, they can be called an '-ing word. Such a word, depending on its context, can serve either as a noun, and name an action/event as a sort of 'thing', or as an adjective, and refer to an action/event as a property or quality of an agent.

If we concentrate on the particular usage of the -ing word itself, taken in the context itself, we find that in some cases it has to be nominal, in some cases it has to be adjectival and in some it could be either and makes no difference to the meaning which. Once we pin it down as a participle or a gerund, we get into needless difficulty.

Is there no way, then, in which I can give a correct answer to your question? Does "it does not matter" lack the necessary rigour? I think not. And there is a way in which to can introduce a grammatical concept taken, this time from ancient Greek: enclitic. For ancient Greek grammar, which is well explained by Herbert Weir Smyth (http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0007%3Apart%3D1%3Achapter%3D7%3Asection%3D12) Essentially, some words in Greek 'lean back ('incline') on an immediately preceding word. And I think that -ing words often do something rather like that. So in your examples, the insertion of the preposition 'by' makes the -ing word 'lean back on it' and so 'shape-shift' into being a nominal use, in turn rendering the whole phrase adverbial. The removal of the preposition causes the -ing word to 'lean back on' the (now nearer) subject of the main verb and agree with it adjectivally. But the meaning is the same.

If I now look at all your sentences as you wrote them, I think that this explanation works. It is all a matter of what comes before this-ing word and the subject. It wants, if it can, to 'lean back' on a noun. If there are more than one, then it (or rather the reader) has to choose which and does so on some basis or other. All subject and object nouns exert some 'gravity' on '-ing' words that follow them. That is why, "I watched them looking out of my window.", in the absence of a context, is awkward and ambiguous. In this case, punctuation would help by inserting a comma after "them". This places a barrier between "looking" and "them", which otherwise would try to 'lean back' and attach to "them", forcing "them" back to the subject "I".

Your sentence 1 looks like the easiest case. But just change it a little and it is less clear.

1a I paid for him using my credit card. ( either "and so we are now evens"; or " and I am now bankrupt and homeless").

This is now ambiguous in the same way as example 2. But Re-inserting the preposition'by' blocks the attachment to 'him').