We find in the Srimad Bhagavatam two explicit references to sadhakas attaining spiritual bodies, one given to Narada and one wherein Dhruva’s material body transformed into a spiritual one. There is also a third veiled reference to the gopis who were restrained from joining the rasa lila, but the other two direct reference are more pertinent to your question.

Your first three questions are related to Dhruva’s case, and Visvanatha Cakravarti Thakura has written an extensive commentary on this section of the Bhagavatam. Therein he stresses the reality of the gradual spiritualization of the practitioner’s body. This phenomena is also mentioned in Sri Caitanya-caritamrita, wherein Sri Caitanya tells Sanatana Goswami that at the time of initiation the practitioner’s body begins to transform into a spiritual body. When this transformation is complete, the practitioner’s body is considered sacred.
After all, once the practitioner’s prarabdha karma is completely eradicated and bhava appears in one’s heart and one’s internal siddha deha suitable for serving in Krsna’s nitya lila begins to manifest, one’s external practitioner’s body is no longer made of karma. It is under the influence of Krsna’s internal sakti. As Sri Krsna explains in the Gita mahatmanas tu mam partha davim prakritim asrita, “Mahatmas come under the shelter of my internal shatki.” This is why we hold sacred sadhus practitoner. Sri Caitanya showed this example by proclaiming the body of Haridasa Thakura, who otherwise appeared to be a so called outcaste, to be sacred.

In answer to your final question, it is best to engage one’s practitioner body such that the fruit of diska is realized, and in proportion to one’s success one’s inner spiritual form for serving Krsna lila also gradually develops at the same time (as it did for Narada). And to the extent to which it manifests by Krsna’s grace, one can cultivate the bhava that it arises out of.

Is there any difference between the service rendered by a siddha in a spiritual body and a siddha in a physical body?

Is there a difference between a siddha doing pracara or arcana externally and internally serving as a gopi or gopi? Of course there is, but in what form the seva takes, not in essence. In either case the desire is to please Krsna in a spirit of saranagati.

Should we aspire for a spiritual body or for spiritualizing the material body to function spiritually?

Quite possibly both, considering that the human body is a medium whichfrom the spirit soul may aspire to associate with the Lord in the latter’s human-like pastimes.

Is the antaryami here Paramatama or Bhagavan? Generally antaryami is synonymous with paramatama.