The office of bishop is an office of spiritual authority, and can be occupied by someone who has not received episcopal orders.  As you note, there are occasionally even popes who were elected prior to the reception of episcopal orders (one in fact-- a Pius, can't remember which, but he was a Piccolomini-- died before ever receiving holy orders, and was never more than a deacon).
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Of course, without episcopal orders, there are certain functions a man cannot perform-- confirmation and holy orders, for instance.  But the authority of an episcopal office can be held and exercised without episcopal orders.

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It is a perfectly fair request.  The Church's canon law renders the matter quite clear, I think:
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Canon 331 §1: The requisites of a candidate for the episcopate are: (1) he must be born of legitimate wedlock. Those legitimized by subsequent marriage are also excluded; (2) he must be at least thirty years of age; (3) he must have been ordained priest for at least five years; (4) he must possess good character, piety, zeal for souls, prudence, and other qualifications to govern the diocese in question; (5) he should have obtained the degree of doctor or licentiate in theology or Canon Law from a school approved by the Holy See, or must at least be well versed in these sciences" (Woywod, trans. Smith, p. 134-35, §242).
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Canon 333: Unless prevented by legitimate impediment, the person promoted to the episcopate, even though he be a cardinal, must within three months from the receipt of the Apostolic Letters [i.e., the papal mandate which appointed him to a diocesan office] receive the consecration and go this diocese within at least four months (Woywod trans. Smith, vol. 1, p. 135, §244).
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Canon 2398: If, on his promotion to the episcopal dignity, anyone neglects... to receive the consecration for another three months, he is automatically deprived of the episcopate (Woywod trans. Smith, vol 2, p. 577, §2254).
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So: the legal requirements the Holy See insists on for candidate eligibility do not include episcopal consecration; in fact, there is a law that anticipates that a man appointed to an office will not be consecrated (given the prescription that he be consecrated within four months); and if he is not consecrated in the prescribed time without a good reason, he is deprived of the episcopate, which obviously could not happen unless he actually had it upon his promotion.
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N.B.: these are obviously disciplinary laws which can change (and in fact have changed; the author of the book from which I am transcribing says the previous law did not insist upon the necessity of priestly orders for appointments, while the current one does).  However, the question it hand is a very simple one: can the Church appoint a man to the episcopacy who does not have episcopal orders? The answer to that question is yes.  
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I do not think that women can be appointed to offices even theoretically, the reason being that in order to hold any office, one must be a cleric (which, in the measure of canon law, means receiving at least tonsure if not greater).  Women cannot be any grade of cleric, not even tonsured, ergo women cannot hold any office.

I thought that when a layman was chosen to be a bishop, he first had to be ordained a priest, then consecrated a bishop.  It would be theoretically possible to rise from the lay state to the episcopate in a single day, but it's been my understanding that he had to be a simple priest first, however short that interval might be.  Not sure if he has to be ordained a deacon before being ordained a priest.

I do know of one case, Bishop Thomas Sebastian, who IIRC became a priest and then a bishop in the same day.  Not sure how canonical this was, just stating the fact.

Tylenda, J. N. (2003). Saints and Feasts of the Liturgical Year (pp. 269–270). Washington, D.C.: Georgetown University Press.

One would assume that someone would have to be ordained a deacon first (as it is one of the major orders), than a priest, than consecrated a bishop.  

This is the common opinion of theologians, but there are some who disagree, arguing that the priesthood is contained within the episcopacy just as the diaconate is contained within the priesthood.  But that is the minority opinion, so a man consecrated without first having been ordained would be considered doubtful.  That is why when you hear of a priest being consecrated a bishop through some valid line while having been ordained by a doubtful line, the bishop/priest must be considered positively doubtful.  There are a few such in the Thuc lineage, where the episcopal consecration was done by one of the clearly valid lines but the man had first been dubiously ordained.