As a Jewish woman, would Mary the mother of Jesus have been able to read the Hebrew scriptures? If not, how did she know Hannah's song and turn it into the Magnificat? Had she memorized it orally?
Excellent question. Though the literacy rate in the first-century Greco-Roman world was only 10%, it was apparently a bit higher among Jews. Since Aramaic was Mary's spoken language, she may have known the Aramaic rendering of portions of the Hebrew scriptures (called targums), or she may have learned Hannah's song by rote memory.
Learning by rote was not unusual in early Jewish education. It was, after all, largely an oral culture, and scriptures were read out, then recited, then put to memory. Parents did sometimes have portions of scripture scrolls in the home, but in any case we know from Luke 4 and Mark 6 there were such scrolls in the Nazareth synagogue.
I've heard that Jesus spoke Aramaic, a language that has been lost in great part. What sources or fragments give us a clue into what Aramaic sounded like? Also, did the Romans who interacted with Jesus speak Latin, Greek, or something else?
In the first place, portions of Daniel in the Old Testament were written in Aramaic. In the second place, we have various ancient Jewish targums written in Aramaic, and of course we have some Aramaic words and phrases in the New Testament, especially in the synoptic Gospels and Acts.
As to what the language sounded like, it's hard to say this far removed in time. Few Romans interacted with Jesus, other than perhaps a centurion and, of course, Pilate. They would have had to speak Greek to Jesus, as they would not have known Aramaic, and it is highly unlikely Jesus knew Latin (which was not a spoken language in the eastern empire, except among Romans). Israel was a mixed-language environment, with the one common language shared by most being Greek.
Is there anything in the Bible stating the names of the days of the week, such as Monday or Tuesday?
No, there is nothing quite like that in the Bible. There are, of course, references to the Sabbath and the Lord's Day, also called the first day of the week. Our modern names for the days come from a variety of sources, some ancient, but none biblical.
According to ancient property laws, it was the first-born son who received the largest inheritance and had the greatest responsibility for continuing and taking on the mantle of family support. Because women were not usually allowed to inherit in many ancient near eastern settings, the first-born son was crucial. If one had no son, the family lost their property.
Were dogs pets in Jesus' day?
There were dogs in Jesus' day and culture, though there is little evidence that they were domesticated. However, the story of the Syrophonecian woman (in Mark 7.24-30) may suggest there were some who had puppies in the house.
How do you explain the parable of the men who worked one hour and got paid the same as those who had worked a full day? Why wasn't the owner of the vineyard more generous to the men who worked all day?
This parable is about the kingdom of God, and about God's undeserved favor or benefit. The parable makes the point that God is fair to all, and often more than fair. It also makes evident that we should not be envious when God is generous to others.
When the men with the talents invested them, what would that have been like? How did you invest money in those days? In biblical times, there was no free market economy or capitalism as we know it; people generally traded on a barter system. However, during the New Testament era a system by which people exchanged money for goods became more common. The system included the loaning of money at interest. Investing in antiquity could involve buying property or a business of some sort.