No, "coming" is not a gerund in this particular sentence fragment though "easing" is.
Present participles can be confusing. They can be used as the substantive part of a verbal phrase as in
I was washing the dishes.
The present participle "washing" in this explains what was going on.
A present participle can be used as an adjective to modify a noun.
The word "avoid" is functioning as the infinitive of a verb and not as a noun. The preposition "to" goes before (or more generally is associated with) a noun. As the word "avoid" is not a noun, the word "to" cannot be a preposition.
You teach someone to do something.
He is teaching you to speak English.
to here refers to a purpose, a reason.
If you teach something, you can say:
I teach English.
I teach sailing. [the activity of sailing a boat]
I am teaching them to speak English.
However, we would not say: I am teaching speaking unless it refers to public speaking, the activity.
For what I assume is the intended sense, OP's example #1 is the most natural alternative:
1) I teach you [how] to speak English
But the second version doesn't have that meaning:
2) I teach you speaking English.
In principle that could make sense if we interpret it as meaning I speak in English while teaching you [something, not necessarily how you ...
The two highlighted sentences are unrelated, so I don't think you can derive a definite common rule for both of them. They do both build on the context provided by previous sentences, which is why they can be understood by the casual listener—they could not reasonably be understood, if they were written on their own.
If you're looking for an explicit ...
(Edit) Ok, so you don't want to make the gerund/infinitive passive, you just want to make the sentence passive. Got it.
The passive voice sounds awkward in this context, and would not normally be something a native speaker would say. Still, if you are determined to do so, the rule is generally the same. Invert subject and object, and change the verb to ...
The passive forms would be as follows:
TV watching is enjoyed by me.
Football playing is not liked by me.
But they sound so unusual that, outside of a mental exercise, nobody would ever actually use them.
There is no direct passive for the additional sentences. (Although a similar kind of translation might be made, it wouldn't be exact.)
The marked phrase is a sentence fragment. You shouldn't be troubled to find it in a script, as it happens frequently in speech. People don't always speak in complete sentences. (Know what I mean?)
The simplest fix would be to simply add a predicate to the fragment, making it a complete sentence:
My dreams are coming true. You and me are saving my lady ...