If works don't earn our salvation, but SHOW our salvation -- then would the "sinner's prayer to receive Jesus" actually be a work showing a salvation that has already taken place, as opposed to a work that earns it?
Yes. But I think most people think of the "sinner's prayer" as the first step in a person's salvation (not something they pray as a result of their salvation).
Since there were many people who wrote books for the Bible but didn't get in, Who decided what books were going to be in the Bible? Are we able to read the books that didn't get in?
The answer to your question has lots of background (thus making it longer than you probably want). So let me first suggest a book for you to read. The book is Why Trust the Bible by Greg Gilbert. It's short and gets to the heart of your question in an accessible way.
The short answer is that there was no "who" that decided which books were going to be in the Bible -- other than God. The Bible came together as a process over many years. For example, and somewhat obvious, the Old Testament was formed much earlier (hundreds of years earlier) than the New Testament. Eventually, the Old and New Testaments were put together. And there are some view the final "stamp of approval" for what is our current New Testament to be in the 4th century (although that's a bit simplistic in my view as that may have been a somewhat "official stamp of approval" but churches had been using the same Bible for many years prior).
Regarding reading other books, sure you can read them. Just understand that they are not in the Bible for a reason (some contain error, some have bad theology, etc...). They may help you understand what other kinds of thinking was going on during a certain time period, which may be helpful. But let me say, read your Bible first. Read it often. Read it a lot. Then go to other writings, but not to the neglect of your Bible reading. And always let the Bible filter whatever else you may read.
We had a question in life group: if the Jews are God's chosen people, why do they reject that Jesus is their awaited Savior?
Jesus didn't meet their expectations as to who the Savior (Messiah) would be and what he would do. Thus they rejected him.
Based on the genealogy in Genesis 11, Shem and Abram would have been alive together for 110 years. Is it fair to say that Abram was a pagan who worshipped other gods? Seems like he may have had equal (or greater) influence from his many-greats-grandpa.
I don't know that we can say, with certainty, that Shem and Abram were alive together by counting the years of the genealogy in Genesis 11. I don't want to get deep in the weeds, but it's unfair to impose upon biblical genealogies the same accuracy of our modern day records. Meaning, sometimes they highlight only important figures in a genealogical record, sometimes they're displaying symmetry (each section of a record has the same number of people mentioned), and sometimes we have records containing different names in our Bible. This doesn't mean the Bible contains error, it means we must understand what the author was wanting to show us in a particular genealogical record. Some examples that show these variations in genealogical records would be Exodus 6, Ezra 7, and Matthew 1 (also in Revelation 7 the twelve tribes of Israel is a different list than what is found in the Old Testament).
But even if they were alive together, we have no way of knowing what Shem's influence would have been on Abram (Genesis 10:32 says that there was a "spreading abroad on the earth after the flood" by Noah's descendants; they may have not even known each other). As we also see, in Genesis 11, people were abandoning God to make a name for themselves (tower of Babel). In Shem's long life, he most likely saw many in his family abandon worshipping his God. The fact that Abram was living in a pagan land (a land of darkness) and that his story, unlike Noah's, doesn't mention him being righteous or walking with the Lord (nothing positive is said about Abram to begin his story) suggests that God calls Abram out of a spiritual darkness to spiritual life as much as He calls Abram out of the pagan land of darkness and into what will become the Promised Land.
I believe God was looking for Abram but how do you know that Abram wasn't looking for God? Where does scripture say that? Thanks!
This is based, not on a specific statement about Abram in the Bible, but about what is said about all people. I'd encourage you to listen to this sermon that I preached on Romans 3:9-20.