There are different perspectives. This is an advaita perspective (the belief that all Gods are One.)
What are the three gunas
The three gunas are:
Tamas - this is the quality of laziness, darkness (ignorance) and inertia.
Sri Swami Sivananda on tamas process
Tamas is inertia or inaction or darkness.
Tamas binds man to heedlessness (Pramada), ...
First of all there is not anything like 44 Avatars of Shiva. There are many (actually infinite) Avatars of Lord Shiva. Also as you have asked "What is the need of his avatar if his function is to just destroy the Universe?" So, I'm here describing Avatars along with significance of them:
Incarnations from Vedas:
Vedas talk of many ...
Vishnu and Rudra are Satvic and Tamasic respectively only in view of activities and in nature they are inverse to this. Brahma is always Rajasic. This is explained in Chapter Sixteen: Description of creation in Rudra Samhita Section One of Shiva Purana. (Pg: 253)
Vishnu, Rudra and I (Brahma) represent the Gunas. Siva is free from Gunas. He is the ...
To an extent they are related. Swami Nikhilananda says in his book The Upanishads (V1, p 59):
With reference to His [Saguna Brahman] three activities of creation, preservation, and destruction, Saguna Brahman is known as, respectively, as Brahma, Vishnu, and Rudra, or Siva. These form the Trimurti, the trinity of Hinduism. The influence of rajas is seen ...
"Do the Trimurti (Brahma, Vishnu, Shiva) cleanly map onto the 3 gunas (sattva, rajas, tamas)?"
NO. Not related in anyway.
God is above 3 modes/gunas. Their physical incarnations can contain all 3 modes (like us), but Gods don't get bound by them.
BG 7.13 — These whole universe being bewildered by the nature of these 3 modes/gunas, doesn't know me who ...
"I think It may be possible demigods could get bored as they are not enlightened but what about Trinity? Is there any case which tells trinity got bored?"
Let's assume that the trinity is one or the other way the "Supreme Brahman", which is the ultimate 'form' of the true self (Atma/आत्म).
If we translate,
bored = disinterested (a kind of indifference)
It is said during universal dissolution the universe gets merged into Supreme Brahman.
Now, if Supreme Brahman = Shiva; the sentence can be said as:
It is said during universal dissolution the universe gets merged into Shiva.
if Supreme Brahman =Vishnu;
It is said during universal dissolution the universe gets merged into Vishnu.
similarly if ...
It seems to be a puranic concept
Here they are mentioned in narayana suktam-
tasyāḥ śikhāyā madhye paramātmā vyavasthitaḥ, sa brahma sa śivaḥ sa
hariḥ sendraḥ so'kṣaraḥ paramaḥ svarāṭ.
In the middle of that Flame, the Supreme Self dwells. This (Self) is
Brahma (the Creator), Siva (the Destroyer), Hari (the Protector),
Indra (the Ruler), ...
Acc to Rāvaṇa’s boon of invincibility that he should not be vanquished by Devas, Daityas, Dānavas, Yakṣas, Nāgas, Gandharvas, Suparnas, Siddhas and other beings. He left out humans and animals as he thought they were too weak.
So that's why Śrī Rāma was incarnated as a Human. Śrī Rāma was the most ‘human’ and simplest Avatāra of Viṣṇu. Śrī Rāma, unlike ...
According to this the importance of Bhrigu Temple and Sage Bhrigu doing penance in the city of Ballia is mentioned.
This town is undoubtedly of great antiquity; many
legends bear witness to its historical importance. It is a common
belief that Ballia was founded after the name of Rishi Balmiki,
the first Sanskrit Poet of India.
The Confluence ...
You cannot say the exact reason why they are doing tapasya. But while doing so, they are praying one another in a cycle.
I have read one short story about this in "Shiv Puran". It says,
Once, Mata Parvati got curious about the same thought that why is Lord Shiva always doing tap? To whom is he praying?
So he asked Lord Shiva the same. In reply to that Lord ...