I have come to understand the sovereignty of God in His predestination of believers, but this understanding leads me to more questions. For example, why are Christians called to evangelize if God is in control of who believes? I know you said believers are rewarded for service, but the Bible tells us that we should serve God with no expectation of reward.
Your experience in coming to understand the sovereignty of God in salvation is certainly not unique. Most commonly, the first question a student asks when introduced to the truth of election is the one you raised: why does God's call us to evangelize given the reality of God's election of believers? Furthermore, how do we find motivation to seek the lost if the fate of every person lies in God's hands?
First, God's call for men to serve Him in any capacity has never been a matter of God's need; He doesn't need our involvement or assistance in any work, including in the work of spreading the Gospel. God is capable of delivering the message of the Gospel to anyone in anyway He desires.
For example, when God calls upon us to give money to support a church or to lead a Bible study or to evangelize the Gospel to others, He doesn't make these requests because He depends on our contribution to His work. God owns everything in creation and He is the Teacher of all Truth by His Spirit, and He is the One Who changes hearts and brings faith. What can we add to His work? Nothing.
Nevertheless, God delights to share in His work with us, so that we might gain the blessing of partnering with the Father and growing in our relationship with Him and thereby give Him glory for His work through us. When we obey the Father and join Him in His work, we receive the blessing of maturing in our faith and the encouragement of watching the Father at work through us.
Also, Paul tell us that we are working to please a Master Who is prepared to reward us in eternity for our faithful service. Our reward is not based on results - only God gets the credit for results - but it is based on diligence and faithfulness.
Pastor Armstrong commonly uses the analogy of an Easter Egg hunt to describe the way we should approach evangelism. Like children eager to begin their search in the yard, we should see the world as our opportunity to find "eggs." In any Easter egg hunt, children are confident their father has hidden eggs somewhere in the yard, so they search with enthusiasm and without tiring because they are convinced they will find an egg if only they search long and hard enough. They rush from place to place looking for an egg and hoping to find it before another child gets it first. They know that if they look in one place and find nothing, they must move on to another place because eggs are in the yard somewhere.
Likewise, once we understand the sovereignty of God in salvation, we can understand our world as a yard filled with Easter eggs (i.e., the elect who the Father is bringing to faith). The Father is always at working bringing men and women to faith around us everyday, so we should go out into that world confident we will find believers if only we look with enthusiasm.
Just as the children don't produce Easter eggs by their searching, we can't create believers by ourselves through our evangelizing. We depend upon the Spirit to grant us the success we desire. Nevertheless, the more eagerly we search, the more we will likely "collect" – and at our judgment moment, a full basket of "eggs" will be to our blessing.
Finally, we wonder where you came to the belief that believers shouldn't do anything to serve God with the expectation of a reward? Perhaps you were thinking of this passage where Jesus said:
Assuming this was the passage on your mind, Jesus wasn't speaking concerning eternal rewards in this passage. Rather, He was speaking of earthly rewards. A believer shouldn't hold out expectations for earthly reward and honor when serving Jesus, but we can – and should – expect reward in eternity.
While certainly the Bible says we are to love the Lord, it ALSO says we are to seek eagerly for His approval and His rewards. Perhaps most famously, Paul says:
From the context of 1Cor 9 it will be clear that Paul is not describing our salvation, since men do not "run" to receive it, not does it require "self-control" etc. These references can only describe the rewards that await the believer who serves God earnestly and diligently. Notice that Paul uses terms common to athletes in the Olympic games to describe the attitude of a successful believer: run your race to win the prize. If we are willing to run earthly races to win earthly prizes, Paul says, then surely we should also give our best efforts to serving God so as to win an eternal prize.
There is nothing wrong or inappropriate about looking forward to our eternal inheritance and striving to make the most of it. Certainly, this is a strong motivating factor for our obedience to the Lord, and the Gospels are filled with examples of this reward being held out to us as a key motivator.
Consider Jesus' teaching on reward: