As per Valmiki Ramayana & Mahabharata, Lord Shiva had made a powerful bow to destroy the sacrifice of daksha. However it doesn't mention the name of that bow. So was that bow Pinaka or was it another bow ? Was it Pinaka bow that was broken by Ram? Was the same bow used in destroying Tripura?
Indeed Valmiki Ramayana does not mention the name of the bow that was broken by Sri Rama. But it did mention that the same bow was used to destroy Tripura.
From Valmiki Ramayana, Bala Kanda
Oh, best one among men, out of the two longbows gods gave one to restive Trymbaka, God Shiva for a combat with demon Tripura, and oh, Kakutstha, that bow alone is the annihilator of Tripura, the demon... and you have broken that alone... [1-75-12]
To answer your title question, Vishwakarma had made two bows. One for Lord Vishnu and other for Lord Shiva. See verse 11
These are the two strong and sturdy unsurpassed longbows, well-designed by gods and well-crafted by Vishvakarma, the Divine Architect, and these are very important among all bows and well-worshipped by all worlds... one broken in your, and the other in my hand.
Sri Rama had broken Shiva's bow, while Vishnu's bow was in Parshurama's hand.
Also if you refer to Shiv Purana (Chp 8 of Rudra Samhita, Yudha kanda), you will find description of the chariot, bow, arrows etc used by Lord Shiva in the battle. The following is stated
The divine chariot of lord Śiva consisting of all the worlds was built by Viśvakarman with devoted effort.
It is mentioned lord of mountains (Meru) became the bow, lord of serpents became string and Goddess Saraswati the bell of the bow
And right at the end of the chapter
At the bidding of Brahmā and Viṣṇu the chariot and its adjuncts were created bythe intelligent Viśvakarman.
Thus from two sources we can conclude that the bow used in Tripura battle was made by Vishwakarma.
Now as far as Pinaka is concerned, it is a different bow made by Lord Shiva himself at the time of swallowing Usanas/Shukracharya. From Mahabharata Shanti Parva
Indeed, having ascertained the fact of the wrath of the high-souled Maheswara of superior Yoga-power, the puissant Usanas began to reflect as to whether he should go to Maheswara or fly away or remain where he was. Thinking, with the aid of his severe penances, of the high-souled Mahadeva, Usanas of soul crowned with Yoga-success, placed himself on the point of Mahadeva's lance. The bow-armed Rudra, understanding that Usanas, whose penances had become successful and who had converted himself into the form of pure Knowledge, was staying at the point of his lance (and finding that he was unable to hurl the lance at one who was upon it), bent that weapon with hand. When the fierce-armed and puissant Mahadeva of immeasurable energy had thus bent his lance (into the form of a bow), that weapon came to be called from that time by the name of Pinaka.