Sri Rama did not break the Shiva's bow accidentally.
Sri Rama did not break the Shiva's bow accidentally. The bow itself became soft when it came into the hands of Rama. It is because he was an able kshatriya. He wanted to see the bow, lift the bow and aim the bow due to his interest.
While many thousands of men are witnessing that right-minded Rama the legatee of Raghu stringed the bow effortlessly. [1-67-16]
Further, that dextrous one has stringed that bow with bowstring and started to stretch it up to his ear to examine its tautness, but that glorious one who is foremost among men, Rama, broke that bow medially. [1-67-17]
The bow is not broken by itself because of its oldness or dryness. Not to forget it is divine architect Vishvakarma who made the bow. It was maintained by Janaka perfectly. So, there is no point of being broken due to oldness. The reason why it broke was the strength of Rama alone. See the bold part. It was neither broken in the beginning nor the end but in the middle. It is only possible when the arms of the person who is aiming the bow are stronger.
There is another instance to show the strength of arms of Rama:
Rama and Lakshmana, with two quivers each and two bows in their hands, and making all the ten directions of compass lambent followed Vishvamitra, as if three headed serpents followed Vishvamitra, and with their highly dynamic physiques and spiritedness they look as if like the Ashwin twin gods of unequal charm following Brahma, the Forefather. 1-22-7
Rama is known to carry two quivers full of arrows onto his shoulders and bows in both the hands. This shows the strength of Rama's shoulders.
Dasharatha Rama didn't show his Vishnu form to Parashu Rama.
It is being publicised that Rama showed his universal form to Parashu Rama when the latter arrived at Mithila. But Rama of Ikshvaku did not show the universal form to Bhargava Rama. Showing universal form was not at all mentioned in the Valmiki Ramayana. This story might be part of later versions of Ramayana. It is because Rama was not aware of His divinity. What actually happened was below.
Then, when Rama is ready to take aim with the arrow on that inscrutable longbow, and when the worlds are being rendered as insentient, then that Rama of Jamadagni is rendered vigourless and he stared up at Rama of Dasharatha. [1-76-11]
Rama of Jamadagni is calloused as his vitality is subdued by the radiance of that lotus-petal eyed Rama of Dasharatha, and he spoke to Rama of Dasharatha, slowly and softly. [1-76-12]
I have realized your touch of nature as that of the Immutable Supreme Being, God of Gods, the Exterminator of the demon Madhu, namely Vishnu, by the touch of your handling that bow... oh, enemy-inflamer, blessedness alone betides you...
Above verse shows that Bharagava Rama showed realised the divinity of Rama by himself and there was no universal form.
All of these gods who have come collectively are beholding you and your next move, for you are an unequalled one in you achievements and to whom there is no counter-dueller in conflicts...[1-76-18]
Here, Parashu Rama is trying to finish the unloosening of the arrow from the bow of Vishnu. Becaue all the Devas were watching these two Ramas from the sky. If these are watching for long time, some devata or a danava might know the secret and purpose of the Rama's incarnation.
The purpose of Rama's incarnation is not hiding divinity.
The purpose of the incarnation is slaying the evil minded Ravana. This can be only accomplished by either a human or a Vanara. It was a boon to Rava that he will be only slain by these both and not by anyone else. So, Lord Vishnu incarnated as a human in the dynasty of Ikshvaku. If Rama knew that he was god, then it will not be able to kill Ravana and the purpose of incarnation would be failed. Hence, the devatas didn't reveal the secret even when they got the chance. This being the first one and the other instance is from the Yuddha Kanda.
The mighty eagle Garuda appears in the battle field and heals the wounds of the Vanaras, Rama and Lakshmana caused by the arrows of Indrajit. Rama did not recognise him too. He asks:
Who are you, the one endowed with beauty, having blissful garlands and anointments, wearing clean garments and adorned with divine ornaments? [6-50-44]
To this question Garuda replies
O, Rama! I am your dearest friend Garuda dear as your own breath moving outside I came here for the purpose of helping you, both. [6-50-46]
O, hero! You need not be so curious to know the cause of my friendship. You will know of it, once you have accomplished success in battle. [6-50-57]
So, it was intentional that the secret of divinity was kept secret from Rama. The devatas waited until the main goal of the incarnation was completed i.e., the slaying of Ravana.
All this was planned even before the birth of Rama. Lord Vishnu asks the Devatas regarding the strategy to kill Ravana.
What is the idea to eliminate that demon's chief Ravana, oh, gods, adopting which strategy I will have to eliminate that thorn in the side of sages?" Thus Vishnu asked the gods. [1-16-2]
They reply that only a human can kill him.
Thus, on getting boon from the Forefather Brahma he has become arrogant and torturing the three worlds, and he is even abducting women. As such, oh, enemy destroyer Vishnu, his elimination is envisaged through humans alone." So said gods to Vishnu. [1-16-6b, 7]
On hearing that speech of gods said that way, he that kind-hearted Vishnu then chose Dasharatha as his father in human world. [1-16-8]
So, the main goal of the incarnation of Rama is eliminating Ravana. As soon as the deed is accomplished, the gods descended to the earth and reminded the divinity of Vishnu and his original nature.
Lifting their long arms, their hands decked with ornaments, those excellent gods thereupon, spoke (as follows) to Rama who stood there, making a respectful salutation to them with his folded hands.[6-117-5]
They try to remind him his divinity. But Rama couldn't recollect this and considers himself as a human being. He asks them to remind what he forgot.
अब्रवित्त्रिदशश्रेष्ठान् रामो धर्मभृतां वरः |
आत्मानं मानुषं मन्ये रामं दशरथात्मजम् || 6-117-11
सोऽहं यस्य यतश्चाहं भगवंस्तद्ब्रवीतु मे |
Rama spoke as follows "I think of myself to be a human being, by name Rama, the son of Dasharatha. You, as a gracious Divinity, tell me that which I as such really am like this."
Then the creator of the world Brahma explains Rama about his real nature. He says in a verse:
अब्रवीच्छृणु मे वाक्यं सत्यं सत्यपराक्रम |
भवान्नारायणो देवः श्रीमांश्चक्रायुधः प्रभुः || 6-117-13
एकशृङ्गो वराहस्त्वं भूतभव्यसपत्नजित् |
Brahma said "Listen to my true word, O the truly brave lord! You are the Lord Narayana himself the glorious god, who wields the discus. You are the Divine Boar with a single tusk, the conqueror of your past and future enemies."
So, it is clear that Lord Rama only knew his original form after the battle with Ravana. This is the reason why he was able to recognise the god of death Yama and talked to him normally. He agreed to depart to his own abode as his purpose of incarnation was completed.