What was the Shatrughna (Lord Rama last brother) role or Significance in Ramayana?
There is not much highlighted role of Shatrughna in Ramayana but his small role is very illustrious. The story starts with the birth of four brothers where Lakshamana was devoted to Lord Rama so was Shatrughna to Bharata. It is mentioned in Valmiki Ramayana, Sarga 18.
भरतस्य अपि शत्रुघ्नो लक्ष्मण अवरजो हि सः || १-१८-३२
प्राणैः प्रियतरो नित्यम् तस्य च आसीत् तथा प्रियः |
Lakshmana's younger brother Shatrughna is a dear one to Bharata, like that Bharata too held Shatrughna dearer than his own lives. [1-18-32b, 33a]
Then Shatrughna was married to Shrutakiirti, daughter of Kushidhvaja as mentioned in Valmiki Ramayana, Bala Kanda, Sarga 73.
शत्रुघ्नम् च अपि धर्माअत्मा अब्रवीत् मिथिलेश्वरः || १-७३-३२
श्रुतकीर्तेः महाबाहो पाणिम् गृह्णीष्व पाणिना |
That honourable king of Mithila even said to Shatrughna, "oh, dextrous Shatrughna, take the palm of Shrutakiirti into yours... [1-73-32b, 33a]
सर्वे भवन्तः सौंयाः च सर्वे सुचरित व्रताः || १-७३-३३
पत्नीभिः सन्तु काकुत्स्था मा भूत् कालस्य पर्ययः |
"Oh, Rama, Lakshmana, Bharata, Shatrughna ... you all are tender-hearted, well-behaved and well-intentioned brothers from Kakutstha lineage... without lapse of time take up other ceremonies..." So said Janaka to bridegrooms. [1-73-33b, 34a]
Shatrughna was also devoted to Lord Rama and Lakshamana along with Bharata. When Bharata and Shatrughna returned from their maternal place and heard about the death of Dasharatha and going of Rama, Lakshamana and Seeta to forest, they became very sad and angry. When Shatrughna got to know that Manthara was the main culprit, he punished her. It is mentioned in Valmiki Ramayana, Ayodhya Kanda, Sarga 78.
अत्र यात्राम् समीहन्तम् शत्रुघ्नः लक्ष्मण अनुजः |
भरतम् शोक सम्तप्तम् इदम् वचनम् अब्रवीत् || २-७८-१
Thereafter, Shatrughna the younger brother of Lakshmana spoke the following words to Bharata, who was tormented with grief and who desired to undertake an expedition to Rama.
गतिर् यः सर्व भूतानाम् दुह्खे किम् पुनर् आत्मनः |
स रामः सत्त्व सम्पन्नः स्त्रिया प्रव्राजितः वनम् || २-७८-२
"When that even-minded Rama is a refuge for all beings, how much more to tell about our seeking solace to him in our grief? He has been abandoned to the forest by a woman."
बलवान् वीर्य सम्पन्नो लक्ष्मणो नाम यो अपि असौ |
किम् न मोचयते रामम् कृत्वा अपि पितृ निग्रहम् || २-७८-३
"Why the so-called Lakshmana, the strong and powerful man, even could not rescue Rama from distress by making our father to restrain from doing it?"
While he was talking thus, Manthara came there and then Shatrughna punished her but she was saved by Bharata.
तीव्रम् उत्पादितम् दुह्खम् भ्रातृऋणाम् मे तथा पितुः |
यया सा इयम् नृशंसस्य कर्मणः फलम् अश्नुताम् || २-७८-११
"Let this woman, who created serious misfortune to my brothers and my father, receive the fruit of her cruel act."
In anger, Shatrughna also told some harsh words to Kaikeyi.
स बली बलवत् क्रोधात् गृहीत्वा पुरुष ऋषभः |
कैकेयीम् अभिनिर्भर्त्स्य बभाषे परुषम् वचः || २-७८-१९
Shatrughna, the strong and the eminent man, thus violently seizing her with rage, sharply rebuked Kaikeyi and spoke harsh words to her.
After that Bharata along with Shatrughna went to forest to bring Rama back but didn't succeed. After that Shatrughna played an important role to manage the city of Ayodhya along with Bharata in absence of Rama.
But the most heroic work of Shatrughna was defeating Lavanasura, the nephew of Ravana. It is described in Valmiki Ramayana, Uttara Kanda, Sarga 82.
Thereupon expandiag his bow up to ears, Satrughna, the most accomplished archer, discharged his arrow against the spacious breast of Lavana. And piercing his heat that arrow entered speedily into Rasatala And having entered Rasatala tha arrow, honored by the Devas, again came to the descendant of Raghu. And being pierced by that shaft the night ranger Lavana fell down on earth like a mountain clapped by a thunder bolt.
So, what I get from above verses is Shatrughna was a devoted brother and son, kind hearted husband, knower of Dharma, a good governor and a brave warrior.
Shatrughna had offered to go and battle the wicked and powerful Rakshasa Lavanasura in the thickly forested Madhuvana, present-day Mathura in Uttar Pradesh state, and killed him. God Rama had lovingly asked Shatrughna to become the ruler of Mathura [Ref: Ramayana 7.76; Bhagavata Purana 9.11.13, Devi Bhagavatam 4.20.56; Brahmanda Purana 126.96.36.199].
Later, when God Rama's envoy reached Mathura to inform Shatrughna of His decision to leave Earth, he had immediately joined his brother in Ayodhya. Shatrughna's son Subahu had succeeded him at Mathura and his other son Shrutaghati became the ruler of Vidisha in present-day Madhya Pradesh state.
Along with Bharata and Shatrughna, God Rama took jala-samadhi by entering River Sarayu at present-day Guptar ghat of Ayodhya. God Rama was cremated on the banks of River Sarayu, present-day Swarga-dwara ghat [Ref: Ramayana 7.121-122; Skanda Purana 2.8.6; Brahmanda Purana 188.8.131.52].
Source: 'Essence of the Fifth Veda' by Gaurang Damani pgs 65, 67