Who can validly confer the sacrament of confirmation?

I will first answer your specific questions.

1. Who can validly confer the sacrament of confirmation according to the Roman Catholic Church?

Ordinarily a validly ordained bishop, but it can be delegated to a priest.

2. Is the sacrament of confirmation as practised by Anglicans recognised as valid by the Roman Catholic Church?

No. Chrism can only be validly administered by a validly ordained priest. The Anglican rite has no valid priests, let alone bishops.

To deny this doctrine is a mortal sin indirectly against the faith.

Pope Leo XIII, Apostolicae Curae (also called On the Nullity of Anglican Orders):

Wherefore, strictly adhering, in this matter, to the decrees of the pontiffs, our predecessors, and confirming them most fully, and, as it were, renewing them by our authority, of our own initiative and certain knowledge, we pronounce and declare that ordinations carried out according to the Anglican rite have been, and are, absolutely null and utterly void.

It also doesn't use properly consecrated chrism since they do not have priests.

3. If a priest conferred the sacrament of confirmation without a bishop's approval, I imagine that would be illicit, but would it also be invalid, with the person in question needing to be re-confirmed?

It would be illicit but valid. For validity only the proper matter, form and intention are required and since jurisdiction is not part of neither for this sacrament it is always valid when the appropriate matter and form are observed.

4. What is the impact of this issue on Christians who move between sacramental traditions - Eastern Catholic, Roman Catholic, Eastern Orthodox, Oriental Orthodox, etc, etc - each of which has slightly different views on how the sacrament of confirmation is to be conferred?

Those who are Catholic adhere to the teaching of the Catholic Church, apostates do whatever they please unto their damnation.

Now with regard to your circumstance...

You state:

"I've never been able to stop wondering why the Roman Catholic Church viewed my Anglican confirmation as valid!"

The Catholic Church does not view your confirmation as valid. The view of some heretics who are, unfortunately, leading people to Hell with their wicked teaching cannot overturn Apostolicae Curae which is completely irrevocable:

We decree that these letters and all things contained therein shall not be liable at any time to be impugned or objected to by reason of fault or any other defect whatsoever of subreption or obreption of our intention, but are and shall be always valid and in force and shall be inviolably observed both juridically and otherwise, by all of whatsoever degree and preeminence, declaring null and void anything which, in these matters, may happen to be contrariwise attempted, whether wittingly or unwittingly, by any person whatsoever, by whatsoever authority or pretext, all things to the contrary notwithstanding.

This heretical response you received from your diocese is a product of the abominable practice of ecumenism which has been condemned by the Catholic Church countless times.

For example, Pope Pius XI, Mortalium Animos:

For which reason conventions, meetings and addresses are frequently arranged by these persons, at which a large number of listeners are present, and at which all without distinction are invited to join in the discussion, both infidels of every kind, and Christians, even those who have unhappily fallen away from Christ or who with obstinacy and pertinacity deny His divine nature and mission. Certainly such attempts can nowise be approved by Catholics, founded as they are on that false opinion which considers all religions to be more or less good and praiseworthy... Not only are those who hold this opinion in error and deceived, but also in distorting the idea of true religion they reject it,... from which it clearly follows that one who supports those who hold these theories and attempt to realize them, is altogether abandoning the divinely revealed religion.

(...)

So, Venerable Brethren, it is clear why this Apostolic See has never allowed its subjects to take part in the assemblies of non-Catholics...

Which brings us to the crux of the matter. Because what you perceive is the Catholic Church is in actual fact NOT the Catholic Church but the prophesied end times counter-Church is why it appeared to you that the Catholic Church's teaching on the validity of Anglican ordinations was contradictory or inconsistent. There are innumerable ways to prove this fact but since that is out of scope for this answer I will merely refer you to this video.