You need to restart the process of acquiring ritual spells via this mechanic.
The process of copying the spell into your ritual book takes 2 hours
per level of the spell, and costs 50 gp per level.
The feat mentions this.
When you choose this feat, you acquire a ritual book holding two
1st-level spells of your choice
This occurs only when you choose the feat.
Afterwards the content of the book, a physical object, is governed by the process described above. So if you lose it, your only recourse is to find more spells and use the rule above to scribe rituals into a new book.
As a general comment, the gist of the game is for the referee to present a setting for the player to experience as their character. In this sense it is a pen & paper virtual reality. This relates to your question in that given the lack of specific rules, you should consider the issue as if you were really there. The Warlock getting his Book of Shadows (PHB 108) is an example of a specific exception to the assumption of a pen & paper virtual reality. In this case the character has a physical book filled with ritual spells. Lose the book and you lose the spells and have to spend time and money to make a new one.
Note that there appears to be no initial cost assigned to making a new ritual book. However, in the equipment list a spellbooks cost 50 gp, and a blank book 25 gp. Personally, I would charge a PC the spellbook cost for a new ritual book. However, if you are not comfortable with that then charge the player the cost of a book. Other than that, there is nothing to prohibit characters from spending additional time and money to make a backup book.
Also, this is consistent with how the Wizard spell book works on page 114, which is the closet relevant mechanic.
Of interest is that the ritual feat is the only way for a sorcerer to cast ritual spells, as he doesn't possess a ritual casting feature as a function of class as far as I can tell.
Yes
Performing a ritual is still casting a spell (you even have to take the Cast a Spell action every turn). Since the Wild Magic Surge feature does not limit it to "when you cast a sorcerer spell with a 1st or higher level spell slot", any form of casting a sorcerer spell can trigger it.
Best Answer
It is not possible without first transcribing it
Unfortunately, the Ritual Caster feat (PHB, p. 169) has only two ways to put spells in the book.
First is when you select the feat:
So those first two spells just automatically show up in the book. The next bullet point describes how to add new spells:
This allows you to transcribe a written spell, either from a scroll or spellbook, into your ritual book.
There is no mechanic for transferring a 'spell known', but not written, into the ritual book. Unless your DM allows creation of spell scrolls by PCs (which they could using the "Scribing a Spell Scroll" (XGtE, p. 133) or the "Crafting a Magic Item" (DMG, p. 128) downtime activity options), then there is no way to transfer a known spell from a Sorcerer(or any other spellcaster) into the spellbook without first putting it in written form.