The person performing the Baptism does not have to be ordained or even Christian. Any person, male or female can perform a valid Baptism.
Yes, but
only in an emergency. (Otherwise, in the Catholic Church, it must be performed by a priest or deacon.)
It is most properly performed by a person who has been ordained
I think the word you’re looking for here is ‘licitly’, not ‘properly’.
However, you seem to be talking about
Catholic baptism. The topic under consideration here is a non-Catholic Christian baptism. Different standard, there.
, but literally anyone can perform a valid Baptism provided they do it in proper form and with intent.
I don’t think you’re saying this (in fact, I hope you’re not!), but it sounds like you’re claiming that you can walk up to your parish’s baptismal font after Mass this Sunday and baptize someone and claim it was ‘proper’. Are you?
The question here is whether the form was proper because the person reciting the formula was not the one immersing.
Thought experiment for you: if you go to confession to a priest who only speaks Chinese, and you use an interpreter in order to confess your sins, is the absolution invalid simply because the priest heard your sins through an intermediary, and not himself?
In any case, the ritual as laid out in the “Rite of Christian Initiation of Adults”, in its general introduction, mentions that it is permitted for the baptizing minister to have others (ordained or lay) assisting him in the immersion. In the actual text of the ritual of baptism, it mentions that “the celebrant immerses” the catechumen. However, it points out that, especially if there are many catechumens, assisting minister(s) may immerse.
So, at the very least, we already know the answer to the general question: it
is licit for an assisting minister to immerse while the minister of the sacrament says the words. To the specific question, it would seem to be reasonable to ask the diocesan liturgy office, but their answer would seem only to be whether they feel it was warranted in this case, especially given that it occurred outside the Church.