If Rudram, Namakam & chamakam are related to Shiva, why they are chanted for abhishekam of other gods?
Are there any other prominent alternative hymns for abhishekam?
Are they chanted only in shiva-agama or it is same in vishnu-agama?
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First of all there is a Panchayatan (formation for offering puja), in this we perform Abhishek to the Aaradhya (main) god for which we are offering puja.
There are following possibilities for Aradhya and other god's
Any Male God(Except Shiva) we use Purusha Sukta
Lord Narayana(Vishnu) and all his 24 avatar's we use Vishnu Sukta (notice we can also use Purusha Sukta instead) each 11 times for Abhishek.
Lord Shiva we use Rudram also known Rudradhya. The Rudradyaya can be done 11 times it makes one Laghu Rudra
All Devi's (female Godess) we use, Shri Sukta, also Lakshmi Sukta can be used for special pujas like Deepawali Pujan.
While doing Navgraha Puja and Performing Abhishekam we use Saura Sukta for Sun God.
For Ganpati or Ganesha people perform Abhishek by reciting Ganpati Atharvasirsha which is widely considered as an interpolation.
(A general Note : All These Suktas' are Derived from Vedas' and has to be sung like the vedic hymns , with full knowledge of how vedic hynms or mantras are chanted)
This has more details on Panchayatan.
Devi Laxmi is then attended with Sri Suktam and other preferred stotras.
I will give an answer based on Vaishnava texts since you are interested in deities other than Shiva. Śrīmad Bhāgavatam 11.27.31 lists some hymns to be chanted during bathing the deity, "svarṇa-gharmānuvākena mahāpuruṣa-vidyayā pauruṣeṇāpi sūktena sāmabhī rājanādibhiḥ". The hymns are called the svarṇa-gharmānuvāka, mahāpuruṣa-vidya, puruṣa-sūkta and rājanā. The svarṇa-gharmānuvāka comes from the Taittirīya Āraṇyaka 3.11.1. The text is given in this answer: What does the Suvarna Gharma Anuvaka of the Taittiriya Aranyaka mean? Mahāpuruṣa-vidya is described in Vijayatīrtha's commentary, "mahāpuruṣavidyayā jitaṃ te puṇḍarīkākṣetyādikayā pauruṣeṇa". This refers to the Jitante Stotram, which begins with the same words. Jitante Stotram is also recommended in Jayākhyasaṃhitā 13.222-223. Puruṣa-sūkta is very well-known, and from Ṛgveda 10.90. Rājanā comes from Kauthumasaṃhitā #318 and Ṛgveda 7.27.1.
Vijayatīrtha's commentary lists some extra hymns that can be added, "svasya suktena viṣṇusūktena... sūktaṃ samudrādūmiriti pāvīryāyetyādisūktaṃ stūyamānaṃ sūktaṃ nīrājanam". That means, the Viṣṇusūkta (Ṛgveda 1.154), Samudrasūkta (Ṛgveda 4.58) and Pavamānasukta (Taittirīyasaṃhitā 5.6.1) can also be sung.