What does it mean if you say you believe in him, but you don't keep his commandments? Or if you say you believe, but his word is not in you?
The key in understanding Scripture is what one believes, not what one says. If we keep the "says" out of it, we can more easily answer the questions.
So, if one believes in Jesus Christ for eternal life, they are saved. No argument. But, if the believer doesn't keep His commandments? He is a disobedient believer. There are many believers who haven't grown up in their salvation (no spiritual growth), so it should be obvious that God's Word isn't in them.
God's Word is only in those who study it and take it regularly. Many don't, unfortunately.
Sure the gifts and the call of God are irrevocable.
Ccontextually, Paul had already described and defined what he meant by God's gifts before he wrote Rom 11:29. So we know what he meant in that verse by considering what he described/defined as God's gifts:
1:11 is about spiritual gifts, which are irrevocable.
3:24, 5:15,16,17 is about justification, which is irrevocable.
6:23 is about eternal life, which is irrevocable.
And, there are no verses that teach that any of these are revocable.
God called the Israelites his people. Does that mean he didn't destroy them in the desert when they went after false gods?
Yes, God called the Israelites His people. And that calling is irrevocable. And, yes, He destroyed the entire first generation of the Exodus except 2; Joshua and Caleb. Even Moses didn't make it to the promised land. Yet, that in no way revoked God's calling the Israelites His people.
Jude 1:5 God gave them the prophets. Does that mean they did not stone the prophets?
I don't understand how this question relates to the statement before it.
So the LORD led them out of captivity. He gave them prophets; So how did they repay him? Jeremiah 18:15
OK. And the point?
First you have to believe God exists, and you have to believe in eternal life, prophets and prophecy and so on. Prophecy is a gift. So if you believe what is true, not what is not true, then you have passed from death to life as Jesus said, 'Truly, truly, I say to you, he who hears my word and believes him who sent me, has eternal life; he does not come into judgment, but has passed from death to life'. John 5:24
I'm not going to argue that prophecy is a gift, but where does the Bible call it that?
But what if a man falls into unbelief?
Since the gift of God is eternal life, and God's gifts are irrevocable, the apostate is possesses eternal life.
Jesus said salt that has lost its saltness is thrown out Mt. 5:13, Mrk. 9:50 and 'he is cast forth as a branch and withers. Branches are gathered and burned. John 15:6
Clearly, Jesus isn't speaking literally here, but figuratively. He used a metaphor reflecting the agricultural economy of the day. When branches aren't productive, they are unusable and cast off. There is no reason to assume this means loss of salvation, especially since we know that eternal life is a gift of God that is irrevocable.
Casting off a branch in figurative language doesn't "trump" Paul's teaching about God's gifts being irrevocable.
Jesus was teaching that those who aren't productive will not be used by God. In Jesus' day, the general consensus by the Jews was that since they were His chosen people, He was using them for His purposes. It was a shock to hear that God would cast them aside, as we see He did for the entire people, and turned to the Gentiles. Remember that Jesus was speaking to very orthodox Jews. For them to think that God would cast them off and use Gentiles, whom they referred to as 'dogs', was quite shocking to them.
So, burning branches and throwing out unusable salt has nothing to do with salvation, but everything to do with being useful or unusable to God. The Jews prided themselves on their election and couldn't imagine that God would ever cast them off and use Gentiles for anything.