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I'm a neutral party in this discussion but here are my answers.  Correct me if I'm off somewhere.

1) One who is ipso facto excommunicated (i.e. not formally, publicly excommunicated) for heresy, schism, apostate etc., is no longer a member of the Catholic Church, therefore not a Catholic.
Yes or No?

I would say they are still members of the Church because it's not a formal, public condemnation but a spiritual one (for lack of a better term).  If they recant their heresy before any formal condemnation, I don't think they have to be re-baptized, so how could they be outside of the Church?

2) Only Catholics are permitted to receive the sacrament of Penance (Confession).
Yes or No?

Yes.

3) Only a Catholic (priest or bishop etc.) in the sacrament of Penance can absolve the penitent.
Yes or No?

Yes.

4) In an emergency, an apostate, heretical, schismatic, ipso facto  excommunicated priest can both absolve and have his sins absolved - and lay people can have their sins absolved from the censured priest in the sacrament of penance.  
Yes or No?

Yes, for lay people, since the Church will supply jurisdiction to the excommunicated priest because salvation is the ultimate end of the Church.  As far as the excommunicated cleric receiving absolution, I don't know, but I would say 'yes', as, following the same logic, the Church wills all men to be saved and will allow absolution, in case of death.

5) Non-Catholics are not permitted to even approach or administer any of the sacraments, including the sacrament of penance?
Yes or No?

Yes.

I assume your logic is that if a non-public, non-formal excommunication DOES remit membership in the Church, then the Church's canon laws mentioning such excommunicated clerics, and their association with confession, do not makes sense.  Ergo, non-formal excommunication DOES NOT remit Church membership.  I agree.

Also, if membership was remitted, wouldn't Baptism have to be re-administered to excommunicants?  I've never heard that it has been.  Or maybe their desire for re-admittance is like BOD?  (ha ha, sorry, I couldn't resist).

Would ACTUAL, public excommunication remit membership?  Would such a cleric be able to provide confession to a layperson in an emergency?  I'd like to know this.

I'm a neutral party in this discussion but here are my answers.  Correct me if I'm off somewhere.

Quote

1) One who is ipso facto excommunicated (i.e. not formally, publicly excommunicated) for heresy, schism, apostate etc., is no longer a member of the Catholic Church, therefore not a Catholic.
Yes or No?

I would say they are still members of the Church because it's not a formal, public condemnation but a spiritual one (for lack of a better term).  If they recant their heresy before any formal condemnation, I don't think they have to be re-baptized, so how could they be outside of the Church?

Quote

2) Only Catholics are permitted to receive the sacrament of Penance (Confession).
Yes or No?

Yes.

Quote

3) Only a Catholic (priest or bishop etc.) in the sacrament of Penance can absolve the penitent.
Yes or No?

Yes.

Quote

4) In an emergency, an apostate, heretical, schismatic, ipso facto  excommunicated priest can both absolve and have his sins absolved - and lay people can have their sins absolved from the censured priest in the sacrament of penance.  
Yes or No?

Yes, for lay people, since the Church will supply jurisdiction to the excommunicated priest because salvation is the ultimate end of the Church.  As far as the excommunicated cleric receiving absolution, I don't know, but I would say 'yes', as, following the same logic, the Church wills all men to be saved and will allow absolution, in case of death.

Quote

5) Non-Catholics are not permitted to even approach or administer any of the sacraments, including the sacrament of penance?
Yes or No?

Yes.

I assume your logic is that if a non-public, non-formal excommunication DOES remit membership in the Church, then the Church's canon laws mentioning such excommunicated clerics, and their association with confession, do not makes sense.  Ergo, non-formal excommunication DOES NOT remit Church membership.  I agree.

Thank you for answering with clear answers to the clear questions and according to the teaching of the Catholic Church, which of course means that you answered all the questions correctly.

Also, if membership was remitted, wouldn't Baptism have to be re-administered to excommunicants?  I've never heard that it has been.  Or maybe their desire for re-admittance is like BOD?  (ha ha, sorry, I couldn't resist).

Would ACTUAL, public excommunication remit membership?  Would such a cleric be able to provide confession to a layperson in an emergency?  I'd like to know this.

Remember, I underlined that the scenario was in an emergency, that is what this is scenario is all about. The Church teaches that things work a bit differently in an emergency than they do under "normal" circuмstances. So yes, even a formally excommunicated priest or bishop can validly absolve sins in an emergency.  

In normal circuмstances, it depends on the offense. If the person (priest or lay man) had been formally declared as excommunicated, then no, they can never be re-baptized, but like any baptized non-Catholic entering the Church, I think they are supposed to publicly (before a priest, bishop or etc.) abjure their heresies, then go to confession.

"Once Catholic always a Catholic", in its essence, is a thesis professing that:

Membership in the Church coincides with the indelible Mark of Baptism, and essentially the two coincide.

I know you are still learning Catholicism, but the mark of baptism only means that a heretic remains under the jurisdiction of the Catholic Church, but it isn't even a controversy that heretics, such as the Lutherans, are outside the Church nevertheless. This is a fundamental.