In the past, depending on the community, it was given to the woman, in part, and in part to her family. In some cases it wasn't money and it could be something of value. Like a ring or jewelry. Now a days a lot of people use the ring as the thing of value. I.e. the man would pay for the rings.
The three methods mentioned were ruled to be an entire process at a certain point. Two of them is from the Torah and the third is from a ruling from an Mosaic court. Thus, a Jewish wedding could take about a month to go through the process and celebrate the future union. During that time when the contract and the price was paid the woman is considered the man's wife. The whole thing was completed and formalized when they are physically together or even alone together. Thus, the entire process would end with the bride being brought to the husband's house and them being alone together in said place. When they came out together, then it was a completely closed deal.
In ancient times, most Jews were married young. So, there was less of an issue since between 13 to 19 is when marriages took place. In the case, where a Jewish man has had relations with a women, back then a young girl, who was available/unmarried but was not his wife in order for anything to be considered on him it would have had to been a situation where he found doing so and had been warned beforehand. In such a situation where the woman was still living with her parents, then the man would have to pay her father's family.
In those times, it would have been extremely rare for a women to live alone with no family. Yet, if such a situation did happen and only the two of them knew about, in some form, he would considered to be married to her just not in the proper way. If they were living together, it would further be understand that that was his wife but that they had no contract together. Thus, if the divorced there would be issues is there was contract.
In modern times, the entire process normally takes place the night of the wedding. The formal part ending when the husband and wife go off to private room together. When they leave the room together the whole process is considered sealed and normally the celebrations of the wedding continue.
In ancient times, see the above. If a situation happened where a young girl who was living at home was found mutually with a boy/man that she was not married to they would be given the opportunity to marry officially. If the boy, man, etc. agreed and she agreed then an official wedding would held. If either of the girl or the boy/man did not want to marry then the boy/man would have to pay a price to the father of the girl.