This is what you wrote in the OP:  Did you ever find out about the first teacher who took your child into the hallway?  Did this really happen?  If not, then it sounds like your child may have made a blatant lie.  If not, I still wonder whether the truth lies somewhere between what your kids said and what the school is saying.

One teacher removed my child from class to speak with him in the hallway about how good and important it is to make the responses, and another teacher told my other child’s class that whoever made the responses would receive two marbles (my child perceived that this was in response to this teachers’ having discovered that he was not making the responses).

Well, later that story changed from one teacher taking one of my sons out into the hall, to another teacher taking my other son out into the hall.  Then later after continuing my questioning, it turned into “maybe the teacher took me into the hall about something else.” 
So...
But here is another weird aspect I am trying to figure out:
Apparently, my spouse and mother, unbeknownst to me, were saying a Novena to help decide whether the kids should go to the school.  The kids told us this info on day 7 of the Novena, and we immediately made the decision (my spouse later informing me of the Novena, which sealed the deal for me).  Day 9 was the school registration deadline, and because of the curious timing of the “revelations” from our children, the Novena in play, and the “coincidence” of the Novena ending on the registration deadline (combined with the fact that my spouse was suddenly willing to homeschool because of the whole episode, which was not at all their position before this all happened), we thought heaven had answered my spouses prayers, and were at peace with it all.
Now, learning after the fact that it is extremely unlikely our children accurately recounted the matter to us, I am wondering whether heaven had spoken after all.
Could/would Our Lady use mistakes or lies to get us to come to this decision?  That seems blasphemous.  On the other hand, could heaven make good (ie., the decision) come from evil?  I think that answer is yes.
Meanwhile, there are other major things happening in our life, which I will not go into, but I have perceived/wondered/felt that something I am about to do has aroused either the anger of heaven or hell (the latter, I think/Hope), and that this incident is a consequence of my future action, though I could be wrong about that:
Did the devil trick us into making this decision, or did heaven respond, or neither?
All I can say is that I have been relieved I will not have to worry about the kids at the school, and had a great feeling of peace after making the decision, even though this subsequent revelation of inaccuracies somewhat muddies those waters.
I wanted them out anyway, so maybe the peace was purely natural, and not the inspiration of the good angel (or, maybe it is the disturbance of soul caused by the devil who wants me to second guess myself which has muddied the waters).  Both possibilities are discussed in the discernment of spirits.
In any case, the rules also say never to make a decision in a state of turmoil.
Consequently, we will, for now, live with the decision and carry on.

I think the Novena stuff is just coincidence to be honest. Problem is, when people ask God for a sign or guidance, they'll always look for the answers they want to find. Also as significant as the Novena ending on the registration deadline day may sound, I suspect your spouse and mother probably planned the Novena so it would end then.

They both say otherwise, and the idea was my mother’s (who would have no idea regarding deadlines).

I also don't see why it ending on that day couldn't have meant just the opposite: to register

The interpretation my spouse gave it (whose inclination was to register) was the opposite, mostly because of the misinformation reported on the 7th day.
I guess the primary argument in favor of not registering, however, is the discernment of spirits:
If we were being influenced by the devil in disguise as an answer from heaven, then the rules for the discernment of spirits say we should have been in a state of agitation, whereas it is peace (despite some slight temporary muddying of the waters after learning our children had not given us the objective facts).
And more than this, the school has been a constant source of agitation and contention in the family (because of the sspx sellout) which was only ever very begrudgingly consented to in order to lighten the load on the homefront, and as a perceived lesser evil/necessity.  But now there is peace without the school.
Finally, and perhaps this is the biggest sign in favor of receiving an answer to the Novena against enrollment at the school, my spouses sudden resignation and acceptance of homeschooling (albeit gained on the basis of false info, it has remained nonetheless), where before it was absolutely a non-option.
In any case, whatever the truth of the matter is, this is the course we have decided to take, and we are happy with it.

I was in Catholic school before Vatican II.  We never had to give any responses.  The Dialog Mass is one reason why I was never keen on the SSPX.  Yes, I know it's traditional to French Catholics, but it isn't to Americans.  Maybe I would have been more receptive to it if the novus ordo never happened, but that Mass soured me on any participation other than internal.

The Dialogue Mass is only a couple decades older than the NO I believe, at least in terms of widespread implementation. It may be pre-Vatican 2, but Vatican 2 was just the modernist coup de t'at, they had great influence and spread a lot of errors even before the council.

Exactly correct.
In fact, the dialogue Mass has only been "traditional" in France since the death of St. Pius X (i.e., 100 years +/-).
Before that, it was unheard of anywhere in the world.

http://www.traditio.com/tradlib/dialogue.pdf

"We conclude this article at the point where we began.  The Dialogue Mass is nothing more than a liturgical praxis.  Although it may not be Modernist it is undoubtedly MODERN and imbued with the spirit of the age which produced it as Joseph Jungman in Missarum Solmenia frankly admits" ,

“...from the Dialogue Mass the Faithful gain a living knowledge of the actual course of the Mass and so they can follow the Low Mass as well as the solemn Mass with an entirely new understanding.  To have been deprived of such an understanding much longer would not have been tolerable even to the masses in this age of advanced education and enhanced self consciousness.  But what is even more important, now that the Faithful answer the priest and concur in his prayers, sacrifice with him and communicate with him, they become properly conscious for the first time of their dignity as Christians.” (!)

"Even if it is readily conceded that Dialogue Mass is neither Modernist nor heretical this is not to say that it is desirable.  Many practices of the Church in previous centuries were abandoned for good reasons and it is most unwise to revive them now.  Even if there was a liturgy in the early Church which approximated to the Dialogue Mass it is well known that there was also Mass in the vernacular, Communion under two kinds and in the hand, Mass sometimes celebrated facing the people and a married priesthood (even the first Pope was married!).  None of these practices are in themselves against the Faith and were quite legitimate but recent history has proved what dire consequences have ensued when many of them were revived after the Second Vatican Counci"

By Fr. Black, SSPX

I disagree with Fr. Black:
It is true that the dialogue Mass may not be heretical (i.e., it is still the very same Roman missal), but by what fiction does one distinguish between as Mass which is admittedly "modernist praxis" but somehow not "modernist?"
If it is modernist in praxis, then it is modernist in principle.  
The principles of the dialogue Mass are the same as those found in the Novus Ordo: Active participation conceived of only as oral participation; a misconceived "right" of the faithful to "participate" in the sacred functions ("Hey, how come only the priest gets to do stuff?!?!"); following from this, a blurring of th edistinction between the ministerial priesthood, and the priesthood of the faithful; a variety of usages to be decided upon by whatever priest happens to be saying Mass (as with the elimination of the sacred language, this introductin of several usages would be used -and was designed- to fracture the unity of faith by fracturing the unity in worship); etc; etc.

Fr. Black is pulling punches when he should be pouring it on.

In fact, the idea of "participation" itself is suspicious.
Our great-grandparents did not go to church on Sunday to "participate in Mass."
They "went to Mass" or "went to go hear Mass."
Mass was something the PRIEST did, not something we all did.
It doesn't matter that it was always true that, yes, when we join our intentions to that of the priest, we are in that sense participation (even actively, though interiorly).
The modernist liturgical innovators took that reality and made a concrete praxis of it.  
But doing that is out of place (as proven by 1900 years of tradition to the contrary).
PS: add to the modernist principles above that of using the Mass as a catechism or teaching tool, and also that of bending the Mass to the (perceived) needs of the apostolate.
The dialogue Mass has spawned only poisoned, modernist fruit: It gave the go ahead to further experimentation, like the new Holy Week and eventually the new Mass.

To split hairs, we "assist" at Mass, which is to say we assist when we participate with the priest as if at the foot of the cross. We do not participate by giving the responses - that is what the Altar Boys do, they respond for the whole congregation, which is why they should be taught to take extra care to respond very clearly, to strike their breast saying "mea culpa" with purpose, not racing through it.

The congregation participates when they stand, sit, kneel, strike our breast, by making the sign of the cross, bowing our heads etc. at the appropriate times, and by adoring Our Lord at the Consecration and so on. That is how we, the congregation participate, not by diologging, which has no place during the Holy Sacrifice.