Can an informal Letter end as:
1) Yours Loving, John
2) Yours Lovingly, John
3) Your Loving Son, John
I guess the 2nd and the 3rd ones are okay; I'm not sure about the 1st one, please help.
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You can end an informal letter any way you like as this is a matter of preference and possibly etiquette rather than grammar.
However, Yours Loving is unusual and somewhat puzzling. So unless you want to mystify the recipient, it's best avoided.
Also, in all three examples, the word loving would usually be in lower case rather than with a capital L. And your name would be on the line below.
Apart from that, examples 2 & 3 seem fine, assuming that your love for the recipient is what you especially want to convey.
When looking at the first two versions, the most common way of ending a letter with the words you picked would be:
Lovingly yours, John.
Typically, an adjective comes before yours.
The last version is fine. Although, in it and the others, only the first word and any proper nouns should be capitalized.
I agree with the other answers that "Yours Loving" sounds very odd (but that if it's informal you can sign off however you like). However they don't explain the why.
Valedictions such as "Lovingly yours", "Sincerely yours", etc. are all variations on shortening the phrase "I am yours".
"Am" is a verb and so it has to be modified by an adverb: "I am lovingly yours", "I am sincerely yours" etc.
If someone said, "I am loving yours," I would respond, "You are loving my what?" :)
Consider the responses to this very similar question: https://english.stackexchange.com/a/166374/314427