How does being a convert enter into it?

Many, or most, cases today would fall under the conditions of Canon 1098 of the 1917 Code, when the authorized pastors and ordinaries, or a delegate of either, prescribed by Canon's 1095 and 1096 cannot be had without great inconvenience, and it may prudently be foreseen that this will last for a month.   As long as there are no diriment impediments, all that matters for a marriage to be valid under the conditions of c. 1098 is the fact that the conditions of c.1098 truly exist when and where the marriage takes place.  Under such conditions, mixed marriages between a baptized Catholic and a baptized non-Catholic are valid, even if in bad faith they are married by a civil official or non-Catholic minister in a non-Catholic ceremony.

These must be proven and verified, but I was taught they are all pre-Vatican 2 annulment grounds.
Perhaps Pax can corroborate.

* Catholic marries Catholic with canonical form - valid sacramental marriage
* Catholic marries Catholic outside of canonical form (outside of the emergency situation where a priest cannot be had for a month) - invalid

* Catholic marries non-Catholic Christian with canonical form - valid sacramental marriage

* Catholic marries non-Catholic Christian outside of canonical form - invalid
* Catholic marries non-Christian with canonical form (i.e., dispensation) - natural marriage but not sacramental
* Catholic marries non-Christian outside of canonical form - invalid

* Two non-Catholic Christians marry - valid sacramental marriage

* Non-Catholic Christian marries non-Christian - natural marriage but not sacramental

* Two non-Christians marry - natural marriage but not sacramental (but if one spouse becomes a Catholic and the remaining non-Christian spouse will not live peaceably with them, the Catholic may remarry per the Pauline Privilege)

Not clear on what you mean by this.  Are you referring to women who are attractive, who have found someone to marry, and don't want to wait a month?

why in the heck is it so important for people to be able to marry within the space of a month?

that lady whose beauty is not visible to the outward observer, the man who doesn't have the social graces to attract a mate

Our Blessed Lord wills that some people have to wait a long time until they can find someone to marry.  He wills that some people not marry at all

True! And consequently, Our Lord wills that others don't have to wait long to marry. And Church Law provides that it's possible within a month.

I don’t think it’s that simple.  I looked up the canons that Principio mentioned and they had nothing to do with any month time period.  So we would all need to read the actual code in context to get more details.
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I could see the only exception of 1 month being applicable to the following scenarios:  1) the couple lives in an anti-catholic country and normal, diocesan churches don’t exist, or are in hiding for persecution, or are spread out and priests aren’t available often (China, Middle East).  2) same issues as above due to war, or other disasters, 3) the couple is on some remote island or mountain place where the church/priest is not close (ie in the Bible Belt), and maybe the couple can’t travel due to family sickness or some other reason.  So they can get married, to avoid the occasion of sin, and then find a priest to get blessed later.
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In all other cases, the couple is obligated to present themselves before a diocesan priest (or Trad priest) and the normal preparations, classes and such are required.  The only reason the priest/Church makes a couple wait is so She/priest can provide classes and to be a witness that the couple is aware of the sacramental obligations and there is no impediment.  But in crisis situations, the couple vowing before God suffices.  

That’s retarded logic.  No one has the authority to disregard canon law, if they want to save their soul.  That’s why Trad clerics abide by the 1917 code 99% of that time.  There’s certain cases where it doesn’t apply but most codes are still applicable.