How can one reconcile the verses that describe the Servants of the Most Gracious as believers only, with the verses that indicate that all those in the heavens and on earth are Servants of the Most Gracious?

Servitude is of two types: special and general.

The special type: the servitude of love, obedience, and submission. This is exclusive to believers, and people vary in it according to their closeness to God and their obedience to Him. The greatest among people in achieving it are the prophets and messengers, foremost among them Prophet Muhammad, peace be upon him, as mentioned in the Quran in contexts of revelation, the Isra (Night Journey), and preaching. Honor lies in achieving this servitude by demonstrating dependence on God and independence from creation, by combining love for God with fear of Him and hope in Him.

The general type: the servitude of subjugation. No creature is exempt from it. All creation are slaves to God, and His decree and judgment apply to them, as in His saying, the Most High: ﴿There is none in the heavens and earth but that he comes to the Most Merciful as a servant.﴾ (Maryam/93). This type of servitude does not imply favor or honor. Whoever turns away from the special servitude is subjugated by the general servitude.