Traditionally an oath is either a statement of fact or a promise with wording relating to something considered sacred as a sign of verity.
There is no Kaffara on non Baligh child, but if his parents wish to pay a Kaffara, they can do so but it is not obligatory neither on them nor on the child.
This is not an oath but a Promise to Allah which is called is Shariat as عهد . The Kaffara not fulfilling the Promise to Allah is similar to the Kaffara of deliberately breaking fast of the month of Ramadan, which is feeding sixty poor believers or fasting of two months or freeing a slave which is not available now a days.
Kaffara of breaking an Oath يمين or not fulfilling Nathr نَذر (Mannat) is feeding ten poor believers or clothing them and if unable then fasting for three days.
Repenting from a sin is very important, and that's very good that you have sincerely repented from it, but it does not mean you are able to deny having committed it if confronted about it. Taking a false oath is a major sin, and worse than that is swearing a false oath on the holy Quran.
As for saving your honour, you need to weigh out what the situation is, and what kind of consequences there will be for revealing this sin, or whether there will be detremental damage to you, or harm.
It might be advisable that you somehow change the scenario, or do tawriyah, to avoid severe backlash, if that is what the situation will be.
With prayers for your success.
Saving your life and/ or life of others is obligatory. If the only way to save life is taking false oath, then it might be permissible if there are no other Islamic reason against it.
Saving honour has many degrees which must be considered before replying this part of your question. We need to understand the real situation first, then reply can be researched according to Islamic rules.
Kaffara for deliberately breaking an oath كفّارة اليمين is feeding ten ooor believers or clothing them, and if unable to feed or cloth ten poor believers then fasting three days.
The oath to do an action needs many conditions to be valid; e.g. the action should be valid by itself. Oath on an action which is Haraam is invalid e.g. an oath not to talk to his brother or any relative or a believer is invalid.
Oath of the wife is invalid if it clashes with the rights of her husband.
Husband can invalidate the oath of his wife.
Oath of the son or daughter against the wish of their father is invalid.
Breaking the Qasam (Oath) needs Kaffara which is feeding ten poor believers or clothing them. If he is unable to do that then to fast three days. That is one time, which will remove the oath.
You need to :
1. Seek forgiveness for breaking your oath.
2. You need to pay the penalty of breaking the oath which is feeding or clothing ten poor believers or fasting three days.
3. You must refrain from committing that sinful act again.
It is disliked ( Makrouh) to say Wallah ( By Allah) or any word with the same meaning to prove a claim if you believe that the claim is genuine. The Hadeeth says: True oaths destroys the houses.
Obviously, false oath is a major sin and it is called the sinful immersing oath because it immerses the liar inside the hellfire.
If you thought that something happened and said Wallah, then discovered that you were not right, you did not commit a sinful act, but a disliked act. You need to inform every one who heard you that you have misunderstood.
The Qasam which is obligatory to fulfill is when you take the name of Allah in it. If you did not take the name of Allah, then such Qasam is not obligatory and no Kaffara on you if you do not fulfill it.
According to Ayatullah Sistani:
Question: If a person breaks his oath, what is he supposed to do?
Answer: If a person takes an oath that he will perform an act (e.g. that he will fast) or will refrain from doing an act (e.g. that he will not smoke), but does not intentionally act according to his oath, he should give Kaffarah for it, which means he should set a slave free, or should fully feed ten indigent persons, or should provide them with clothes. And if he is not able to perform these acts, he should fast for three consecutive days.
Verse number 89 of Sura al-M'idah has clearly legislated the compensation for a broken oath.
"The atonement for it is to feed ten needy persons with the average food you give to your families, or their clothing, or the freeing of a slave. He who cannot afford] any of these [shall fast for three days. That is the atonement for your oaths when you vow. But keep your oaths. Thus does Allah clarify His signs for you so that you may give thanks." (5: 89)
According o this verse, the compensation of an oath in our time would be to feed ten poor individuals or provide clothing for them. In case this is not affordable, he person must fast for three consecutive days.