For a saṁnyāsin/brahmacārīn (dvija in brahmacaryāśrama), maithuna (with anyone) is niṣiddhakarman.
Seeking services of veśyā (i.e. having maithuna with her, for a śulka) by a non brahmacārin/saṁnyāsin is not an aparādha with an associated daṇḍa, provided the veśyā doesn't belong to a varṇa preceding his own. Such a karman doesn't come under the criteria of saṅgrahaṇa (by taking into account the jīvikā of a veśyā), unless the veśyā is herself in a committed relationship with a man. As Nārada smṛti (12.78-79) states-
svairiṇyabrāhmaṇī veśyā dāsī niṣkāsinī tathā / gamyāḥ syurānulomyena striyo na pratilomataḥ //
āsveva tu bhujiṣyāsu doṣaḥ syāt paradāravat / gamyāsvapi hi nopeyādyattāḥ paraparigrahāḥ //
In similar vein, Vātsyāyana in Kāmasūtra (1.5.2) states,
tadviparīta uttamavarṇāsu paraparigṛhītāsu ca / pratiṣiddho'varavarṇāsvanirvasitāsu / veśyāsu punarbhūṣu ca na śiṣṭo na pratiṣiddhiḥ / sukhārtatvāt //
As Vātsyāyana says, kāmasambandha with veśyā is neither pratiṣiddha nor vihita, because that is purely for ratisukha (not vivāha and santānotpatti). And that there's no doṣa in being veśyākāmī, as per him (veśyākāmino'yamadoṣaḥ - Kāmasūtra 2.9.27). It's important to note that Vātsyāyana is nowhere prescribing/commending taking the services of a veśyā, as becomes clearer in the later adhyāyas too, but is only discussing regarding it. Also, the sixth adhikaraṇa of Kāmasūtra, which contain six adhyāyas, deals with veśyās/gaṇikas primarily. If you are interested in knowing about their description, that would be helpful alongwith Yaśodhara's commentary Jayamaṅgalā. Kauṭilya's Arthaśāstra also mentions important info related to veśyās/gaṇikas, including their business, etc. Samayamātṛkā by Kṣemendra satirizes veśyās and their business. Several Bauddha and Jaina śāstras too mention about veśyās and their business. So, the business of veśyās was actually a pretty established practice in even pre-12th century Bhārata.They weren't seen as unlawful, and the rājya accepted their practice.
However, that doesn't imply śiṣṭas approved of the veśyā-business.
Now, although anuṣṭhāna of such karman (taking services of a veśyā, considering the conditions above) doesn't bestow daṇḍa on the veśyā's client, it's still a pāpakarman on his part, as it makes him an adhikārin of prāyaścitta. This is stated by Nandapaṇḍita in his commentary Vaijayantī on Viṣṇusmṛti (53.7), after he quotes Nārada smṛti (12.78-79)
gamyatvaṁ ca daṇḍābhāvamabhipretya ; na tu prāyaścittābhāvam
Many śāstras state prājāptya as a prāyaścitta for gamana with veśyā
paśuveśyāgamane ca prājāpatyam // (Viṣṇu smṛti 53.7)
paśuveśyābhigamane prājāpatyam vidhīyate // (Atri smṛti 269 ; Mitākṣarā (2.290)
paśuveśyādigamane mahiṣyuṣṭrīkapīstathā /
kharīṁ ca sūkarīṁ gatvā prājāpatyavrataṁ caret //
Nandapaṇḍita's Vaijayantī (53.7) mentions another prāyaścitta, ascribed to Yama
veśyāgamanajaṁ pāpaṁ vyapohanti dvijātayaḥ /
pītvā sakṛt sakṛttaptaṁ saptarātraṁ kuśodakam //
Were there any instances where gods were directly involved in any such thing ?
This is a really ambiguous question put forth by the OP. Is he trying to ask, if devatās were involved in 'prostitution'? If so, the question doesn't make any sense.