A brief comment on the worship using the imagery of human / animal body parts

The worships, using human / animal body part symbols, are anti-Vedic and they violate the rules of Mimamsa (the Purva Mimamsa especially). These worship symbols (using the imagery of human / animal body parts) are based on the fictional puranic tales which have no basis according to the Vedas (Rig Veda etc.*).

Even the Shiv Ling (meaning 'auspicious symbol' in Sanskrit) has no relation to phallus. Shiv Ling, made of stones etc., was created long ago to represent the fire (flame) of yajna. It helped people in offering libations during worship easily and quickly without lighting the actual fire (yajna), since starting / lighting the fire (especially during yajna) by rubbing sticks / stones against each other used to be very difficult and time consuming long ago.

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* Subhash C. Sharma, "Compatibility of a text with the Srutis," Sept. 2, 2006, http://web.archive.org/web/20090809230814/http://geocities.com/lamberdar/sruti_compatibility.html

Please read about the type of questionable Puranic tales in the Appendix in above link ("Compatibility of a text with the Srutis").

Incidentally, without making use of mythical puranic stories etc., the deity images for NarSingh, Ganesha and Hanuman can also be explained according to the artistic creativity originally, just like Shiv Ling (meaning literally as auspicious symbol) depicting the yajna flame / fire in art long ago.

The name / word NarSingh, meaning literally in Sanskrit 'king of men' ('Nar' meaning man / men / people and 'Singh' meaning king or lion), probably was used originally in relation to omnipotence of God and his control over the Creation. This probably later (on the basis of Sanskrit meaning for 'Nar' as man and 'Singh' as lion)led to NarSingh's portrayal in art (to help in worship etc.) jointly as man (body) & lion (head).

The name / word Ganesha (Gana+Isha, meaning literally the 'lord or God of masses' in Sanskrit: Gana meaning masses / people and Isha meaning Lord or God) probably was used for God during the earliest times; Ganesha even today is offered the first worship due perhaps to his earliest beginning. The artist later (for visualization and worship etc.) probably portrayed the abstract Ganesha ('lord of masses') in art according to the similar sounding word 'Ganika' (used for elephant), giving rise to the Ganesha image with elephant head and human body.

Hanuman / Hanumat is used in Sanskrit and Hindi for a strong-jawed man, in addition to monkey. Hanuman in the Ramayana thus probably could have represented a very powerful and big man, belonging probably to a race different from Rama's. Hanuman and his army in the Ramayana therefore might have been a group of individuals belonging to another race of men and not necessarily the monkeys.

Btw, the use of solid Shiv ling or Shiva linga (the auspicious symbol:  Shiv or shiva in Sanskrit originally meaning auspicious and ling or linga meaning symbol) in the place of live fire / flame to assist in worship / yajna also led over time, through the repeated use of the word Shiva in Shiva linga,  to acquire the name Shiv (Shiva) for deity (God / Brahman) during the post-Vedic era.

Note, the word / name Shiva for deity does not exist in the Vedas, even though shiva in the sense of auspicious frequently occurs in the Vedas. This indicates that Shiva (Shiv) as deity evolved during the post-Vedic era, most likely due to the frequent use of shiva (meaning auspicious) in worships using shiva linga or shiv ling (auspicious symbol made of solid material) in the place of live fire / flame (yajna). 

Eventually, after the name Shiva / Shiv for deity (God / Brahman)  evolved and became frequently used during the post-Vedic era, the corresponding literature (Shiva purana etc., comprising many mythical tales and including some Vedic / Upanisadaic philosophical content) appeared in support and glorification of Shiva (Shiv) as deity.

To repeat here again, the names for deity (Brahman or God) in the Vedas and elsewhere appeared first, reflecting usually certain specific divine qualities / attributes which were important to humans / devotees in relating to God. Most of these divine attributes also surpassed the human imagination and characterization, as elaborated also in the Bhagavad Gita: Ch. 10). This was especially true to help in the conceptualization of Brahman during prayer, worship etc., essentially as Saguna (the reference below*, according to Adi Samkara).

The names Agni (the name for Brahman in the terrestrial region), Indra (the name for Brahman in the midair), Savitar or Surya (name for Brahman in the heavens), Ganesha (implying as the lord of the masses), NarSingh (implying as the ruler or as supreme among men), and so on, thus were helpful in easy conceptualization and developing relationship with Brahman during worship etc.

The artist later created the imagery related to deity based on the abstract names and attributes used for Divinity. Eventually, the mythical stories, as part of secondary texts, were added / circulated to support the idea of deity as individual / distinct personality.

Vishnu, in the Rig Veda, is a name used for Brahman in the sense of protector. Moreover, Vishnu is indicated there in the company (having attributes) of Savitar and Indra. This probably led the artist to portray Vishnu anthropomorphically as Chatur-bhuj (having four arms / hands), with two arms / hands of Vishnu's image reflecting the potencies related to Savitar (Sudarshana Chakra or solar disc as the source of light, and kamal or lotus as the sign of life due to sunlight) and other two arms / hands of Vishnu's image reflecting potencies related to Indra (Sankha or conch shell reflecting the sound of thunder of Indra, and Gada or mace reflecting the thunderbolt of Indra).

*Ref. : "Vidya gives the highest positive conceptual account of Brahman by equating it with the attributes of being, consciousness and bliss, which are self-sufficient. Avidya, or lower knowledge, applies attributes which imply relation, such as creatorship and rulership of the universe. These are thus two views of the ultimate, higher and lower. Where, by discarding the differences of name, form, and the like, ascribed by Avidya, Brahman is indicated by negative expressions, as not gross, etc., it is the higher (param). But where, on the contrary, exactly the reality is described, for purposes of worship, as distinguished by some difference or other, it is lower (aparam). Brahman cast through the moulds of logic is Isvara. It is not the highest reality, since it has no meaning for the highest experience where existence and content are no longer separated. Yet it is the best image of the truth possible under our present conditions of knowledge. The Saguna Brahman is not the mere self-projection of the yearning spirit or a floating air-bubble. The gleaming ideal is the way in which the everlasting real appears to our human mind." .. https://web.archive.org/web/20091026134952/http://geocities.com/lamberdar/brahman_absolute.html

Incidentally, the name (title) Devi in the Vedas is used for Brahman (God) in the motherly sense for protection and safety, including praying for relief against diseases and enemies.

The use of puranic and other tales about Devi as deity subsequently, under also the names Durga, Mata and MahaKali et al., thus added to the prayer and worship of Ishwara (Brahman or God as Saguna) in the motherly sense and primarily for seeking protection and safety.

More on the different names for deity in Hinduism in the following: "Brahman (God) in Hinduism," Feb. 24, 2004, http://web.archive.org/web/20090809230806/http://geocities.com/lamberdar/brahman.html