I suppose this is a typical question, but as I understand it, a divorce does not end a marriage. The parties are not free to remarry, but what if after a divorce one of the ex spouses dies? Is the surviving ex free to remarry?

I know I may have worded that confusingly, maybe I can give an example.
Jane and John marry. They divorce. John passes away. Is Jane then free to seek a new marriage?

Also, would sex be considered marriage? The Greek word Agamos used in 1 Cor. 11 is unmarried. Gamos seems to mean marriage. Gamo seems to mean sex…so is sex Biblically marriage?

Last question, is our modern day engagement equal to Biblical betrothal?

These questions may seem silly and maybe they are. I’m just so confused from what I’ve studied. I’m almost convinced there is no answer to these questions. If the answers are too long, we can talk. I’m not sure if the answers are simple and I’m making this harder than it is or not.

I’m not sure it is correct to say “divorce does not end a marriage.”  We are not under the law but even under the law Deuteronomy 24:1,2 says:  “When a man hath taken a wife, and married her, and it come to pass that she find no favour in his eyes, because he hath found some uncleanness in her: then let him write her a bill of divorcement, and give it in her hand, and send her out of his house.  And when she is departed out of his house, she may go and be another man’s wife.”

In his earthly ministry to Israel, under the law of His Kingdom, the Lord restricted divorce only for fornication, which is such a case divorce would end a marriage.

But to the point of your question, yes, death under any situation ends a marriage – Romans 7:1-3.

The Biblical use of the word “marry” is what we would call “consummate” or as you said “sex.”  However even in scripture there is more to marriage than the consummation.  There was always a public announcement or celebration making known to others these two are joining in a holy matrimony.  Otherwise there would be no such thing as fornication, just a whole lot of marriages.  I Corinthians 7:2 says “to avoid fornication, let every man have his own wife.”  So when a woman became a man’s wife they then marry in sanctification and honor; and at the same time everyone knows these two belong to each other.

Biblical betrothal is not the same as modern day engagements.  In Matthew 1:18  “when Mary was espoused to Joseph, before they came together…” in this verse and following (and others concerning them) we see that Mary is already considered to be Joseph’s wife, even to the point that to “put her away” would be to divorce her even though they had not consummated the marriage.  In Bible times, a man and woman would be espoused and considered husband and wife, but waited a year to prove a woman’s virginity.  This certainly is not the same as today’s engagement to be married.  Today a man and woman wed and become husband and wife and consummate the marriage all within a 12 hour period.