There's something I'm confused about regarding civil weddings... if two non Christians get married in a civil ceremony, are they married in the eyes of God or not? if they then divorce and remarry, would they commit adultery or have they been living in fornication the whole time? What if two Christians get married in a civil ceremony? And finally, what if the two non Christians eventually become Christian but never get re-married in the Church? (for example, if they become Catholic, but could be Orthodox too).
Assuming two non-Christians means both are infidels (neither have ever been baptized), according to the Church, they are married. The Church has no jurisdiction over the non-baptized but as the only defender of the Sacrament, She does have jurisdiction over both the Contract and Sacrament of Matrimony. She recognizes the contract as valid though not sacramental for many reasons but in a nutshell, the marriage is recognized for the good of society.
In the case of two Christians getting married in a civil ceremony, the marriage Contract is presumed valid though non-sacramental and if Catholics get married that way, the marriage is still valid though non-sacramental and even sinful or illicit and depending on the situation, things can get quite murky - see a priest.
Some thing to remember is Marriage is made up of two parts, Contract and Sacrament. When vows to remain together "till death" are taken, those vows are what seals the deal on the Contract. Those vows are what God hears, accepts and expects both spouses to remain true too until one of the spouses die, regardless if the marriage was sacramental or not.
It is the Contract that binds, not the Sacrament. Simply put, the purpose of the sacrament is to help the spouses remain true to their vows till death. The Contract needs no priest to be valid as the spouses actually marry themselves, the marriage becomes sacramental when it gets sanctified by the priest - and only by the priest (or other Catholic cleric) - so any marriage entered into outside of Holy Mother the Church, though non-sacramental is still presumed valid as long as both parties were free to marry in the first place.
Because Holy Mother the Church is the only defender of the Sacrament, by default She also necessarily guards the Marriage Contract - which is why a marriage is valid even when not sacramental. Converts who married outside the Church (or even Catholics who did so) seemingly would certainly want to get their marriage sanctified at some point, I do not know, but I don't think it is absolutely necessary.
As for the conversion of one or both infidels, you may want to look up Pauline Privilege as that opens up a whole different can of worms.