I don't have to answer your four questions because none of them are relevant to the issue or in the
context of 1Tim 4:10. I don't waste my time with non sequiturs. You keep avoiding that context of 4:10 and the larger context of the book itself (2:1-7). It's this latter passage that reveals Paul's mindset and what
HE meant by "all men", which Timothy, another Jew, would have understood. And he confirms this mindset in 4:10 with the qualifying clause "especially of those who believe".
Let me lay it out for you one more time a little bit differently and go a little bit deeper into the context as well.
1. in 1:15-16, Paul told Timothy that Christ came into the world to save sinners -- NOT "all" sinners or each and every sinner. Paul goes on to relate to Timothy how God was merciful to him the worst of all sinners so that Christ might display through him his unlimited patience for those [sinners] who would believe on him and receive eternal life. Clearly, "sinners" is qualified in this passage also since Christ's infinite patience extended to his Father's elect, i.e. those who
would actually come to believe on him (See 2Pet 3:8ff. for a very similar teaching.) Also up to this point in his epistle, God sending Christ into the world to save sinners could easily be construed by Timothy to mean Christ came to save only his OC people -- JEWISH sinners.
2. In 1:1 Paul told Timothy that he was an apostle by the command of God
OUR Savior and of Christ Jesus
OUR hope. Timothy, being a fellow Jew, would or could naturally construe this as meaning that God was the Savior and Christ was the hope of his chosen, covenant people Israel. And this is precisely why Paul didn't leave it like that. He wanted Timothy to clearly understand that God's great salvation extended to "all" men -- men other than Jews! Or as the Sovereign Grace expositor pointed out -- that God's salvation extended to ALL KINDS of men -- not just Jews.
3. In 2: 1, Paul urged Timothy that prayers and intercessions and thanksgiving be made for "everyone" -- or more explicitly for kings and and all in authority. So, he's not even urging here that these prayers, etc. be made for everyone in the distributive sense, since he again
qualified his exhortation by limiting "everyone"
to kings and all those in authority. There's no need to take "everyone" in the distributive sense since Paul himself didn't explicitly express himself in that sense. Plus Paul here is getting ready to make his point about Gentiles, since "kings and all those in authority" would be referring to Gentiles.
4. In 2:4-6, Paul reveals the affective will of God (not to be confused with his decretive will) in that He desires for all [kinds of] men to be saved because there's one mediator between God and men who gave himself as a ransom to all [kinds of] men. What kinds of men? Even a cursory reading of the bible will tell us that God always intended to bring his salvation through his OC people the Jews to the
NATIONS of the world. But this is far cry from saying that God always intended to save the entire world in the distributive sense.
5. In 2:7, Paul tells Timothy that what he just got done saying in vv. 4-6, was the very purpose for why God appointed him a herald, an apostle and teacher of the true faith.
But to whom specifically? Paul didn't say to the entire world. Or even to Jews and Gentiles! His mission was to the Gentiles! In Paul's mind "all men" = Gentiles or even more specifically the Gentile Nations. "All men", therefore, can only be understood in the limited sense. The "Gentiles" to whom God sent Paul
qualifies "all men". If Paul meant to say that God is the Savior or all men in the distributive sense, then Paul would have had to include the Jews somewhere in his statement.
6. In 4:10, the
qualifying clause at the end of the text limits the Saviorhood of God to those who believe. It's totally disingenuous and entirely gratuitous to say that God is the actual Savior of all who believe
and at the same time NOT the actual Savior of all unbelievers, but only a
wanna-be Savior. (Is "is" anything we want to make it to be, even when violating the Law of Non Contradiction?)
Now that Paul has clearly revealed to Timothy that the Gentiles have always been within the scope of God's grand scheme of redemptive history, freeing Timothy from any false notion or potential false idea that God is the Savior of the Jews only, he must now also guard his young protege from going to the other extreme with the respect to the Gentiles. God is most definitely not the Savior of unbelieving Gentiles, only of believers.
And this is the reason for the qualifying clause at the end of v.10.
I find no need to go outside the intermediate context of the bible (the book under consideration) when the context makes it crystal clear how we should understand key terms or phrases. Plus you should also do an extensive word useage study of the Gr. term "pas" which is translated "all". You would learn that "pas" is often used in the limited sense.