Were the people of Dawood and Sulaymaan (peace be upon them) a disbelieving people?
Question

I have recently read in a source other than the Qur’an and Sunnah that although Dawood and Sulaymaan (peace be upon them) were monotheists who affirmed the oneness of Allah (Tawheed), the people in their kingdom were disbelievers and polytheists who associated others with Allah (shirk). What is your opinion concerning that? Please answer me with evidence from the Qur’an and saheeh Sunnah. (I am keen to avoid sources from the Israa’eeliyyaat (reports from Jewish sources)).
Praise be to Allah.There is no sound way to prove or deny anything about the earlier prophets (blessings and peace of Allah be upon them) except that which is affirmed or denied by the Holy Qur’an and the saheeh Sunnah of the Prophet (blessings and peace of Allah be upon him). That is because, apart from these two sources, other scriptures that were sent down to previous nations have been distorted, altered and tampered with, so whatever information they contain cannot be trusted. What is mentioned in the Holy Qur’an is definitively sound, and whatever is not mentioned in it, there is no way to prove or deny it; rather we refrain from passing judgement and say: Allah knows best.
Hence the Messenger (blessings and peace of Allah be upon him) said: “Do not believe the People of the Book and do not disbelieve them; rather you should say: We believe in Allah and what has been sent down to us.” Narrated by al-Bukhaari (4485). This has to do with matters that are neither affirmed nor denied in the Holy Qur’an.
If we look at the verses of the Holy Qur’an that speak of Dawood and Sulaymaan (peace be upon them), we will find that they clearly indicate that their people were believers and monotheists who affirmed the oneness of Allah.
In the story of Dawood’s slaying the disbelieving king Jaloot (Goliath), Allah, may He be exalted, says (interpretation of the meaning):
{But those who were certain that they would meet Allah said, “How many a small company has overcome a large company by permission of Allah . And Allah is with the patient”} [al-Baqarah 2:249].
This indicates that the army who fought alongside Dawood (peace be upon him) believed in Allah and affirmed His oneness.
Allah, may He be exalted, says (interpretation of the meaning):
{And We certainly gave David from Us bounty. [We said], “O mountains, repeat [Our] praises with him, and the birds [as well].” And We made pliable for him iron 
 [Commanding him], “Make full coats of mail and calculate [precisely] the links, and work [all of you] righteousness. Indeed I, of what you do, am Seeing.” 
And to Solomon [We subjected] the wind - its morning [journey was that of] a month - and its afternoon [journey was that of] a month, and We made flow for him a spring of [liquid] copper. And among the jinn were those who worked for him by the permission of his Lord. And whoever deviated among them from Our command - We will make him taste of the punishment of the Blaze 
They made for him what he willed of elevated chambers, statues, bowls like reservoirs, and stationary kettles. [We said], “Work, O family of David, in gratitude.” And few of My servants are grateful } [Saba’ 34:10-13].
This indicates that the family of Dawood believed in Allah, worshipped Him and gave thanks to Him. What is meant by the family of Dawood here is Dawood and his children and family." (Tafseer as-Sa‘di, p. 795).
And Allah, may He be exalted, says (interpretation of the meaning):
{And gathered for Solomon were his soldiers of the jinn and men and birds, and they were [marching] in rows} [an-Naml 27:17].
Undoubtedly these groups were obedient to Sulaymaan (peace be upon him), so they must have been believers in Allah. All the verses indicate that.
Allah, may He be exalted, tells us about Sulaymaan (peace be upon him) and his interaction with Balqees and her people:
{[Solomon] said, “O assembly [of jinn], which of you will bring me her throne before they come to me in submission?”} [an-Naml 27:38].
This indicates that his troops were Muslim, because they helped him to bring Balqees and her people to Islam. And Allah, may He be exalted, says (interpretation of the meaning):
{A powerful one from among the jinn said, “I will bring it to you before you rise from your place, and indeed, I am for this [task] strong and trustworthy.” 
Said one who had knowledge from the Scripture, “I will bring it to you before your glance returns to you”} [an-Naml 29:39-40].
It is very clear that they both believed in Allah, may He be exalted. The first one described himself as being strong and trustworthy, and the foundation of trustworthiness is fear of Allah, may He be exalted. Allah, may He be exalted, described the second one as having knowledge of the scripture, which indicates that he was a believer.
And Allah, may He be exalted, tells us that Sulaymaan said (interpretation of the meaning):
{Return to them, for we will surely come to them with soldiers that they will be powerless to encounter, and we will surely expel them therefrom in humiliation, and they will be debased} [an-Naml 27:37].
These troops of his fought in Allah’s cause, and fought against the disbelieving people, so how could they not be believers?
What Allah, may He be exalted, says regarding Balqees – {She said, “My Lord, indeed I have wronged myself, and I submit with Solomon to Allah, Lord of the worlds”} [an-Naml 27:44] – definitively indicates that she submitted to Allah and believed in Him.
All these verses clearly indicate that the people of Dawood and Sulaymaan (peace be upon them) were believers who affirmed the oneness of Allah, so no attention should be paid to anything that says anything different.
And Allah knows best.