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I've always thought of it as 'live' as in 'life', but I learned about TTL first in the context of IP packets where the meaning is probably less ambiguous.

I've had this discussion with multiple colleagues. I say it live as in life. However, most people I know say live as in "until the new record is live"

Is homograph a synonym of homophone or homonym?

technically neither;

homographs are two words written the same way but with different meaning.

homophones are two words that sound the same but written differently.

homonyms are somewhat a combination: written and sound the same, but have different meaning.

since synonym means "having the same meaning" you might get away with considering a homograph a synonym of homonym, but it's best to be precise in these cases.