Does the hadith "None is diligent in maintaining ablution except a believer" apply only to Muslims, or does it include all people, and does the same principle apply to all prophetic hadiths?

There is a difference between doubt and waswasa (whispers/obsessive thoughts). Waswasa is an unsolicited attack on the heart, and disliking it indicates pure faith. Doubt, on the other hand, is the opposite of certainty, such as not being firm in believing the Prophet Muhammad (peace be upon him) to be truthful, and a servant cannot be a believer while entertaining it.

The meaning of the hadith (definition): "No one maintains his wudu' (ablution) except a believer" has several interpretations: 1. Maintaining the renewal of wudu' every time one invalidates it, as mentioned by Al-Iraqi and Al-Mubarakpuri. 2. Maintaining wudu' at its prescribed times, as held by Al-Sindi. 3. Purity of the outward and inward, and maintaining spiritual struggle (mujahada), as interpreted by Al-Manawi. 4. Performing wudu' thoroughly and fulfilling its sunan (practices of the Prophet) and etiquettes, which is another meaning given by Al-Manawi.

The relationship between maintaining wudu' and faith is that wudu' is among the hidden acts of worship that only a believer who observes Allah and seeks reward maintains, especially when there is hardship, such as extreme cold. This is evidence of perfect faith.

If the questioner maintains wudu' according to these meanings, then this is evidence of his faith, and what he complains of is nothing but waswasa, and rejecting it and shunning it is pure faith.