In my understanding, the following sentences are acceptable but I am not 100% confident.
Please give me your opinion.
1. He runs fastest.
(In this case, "fastest" is the superlative form of an adverb, and the definite article before "fastest" is not required though "He runs the fastest," is acceptable as well.)
2. He runs the fastest of all the boys in his class.
(In this case, the definite article before "fastest" is required, because it is followed by "of phrase.")
3. He is the fastest runner.
(In this case, the definite article before "fastest" is required, because "fastest" is an adjective.)
4. She is happiest when she is alone.
(In this case, the definite article before "happiest" is not required, because it is not a sentence comparing her with other people.)
5. This lake is deepest here.
(The similar to the reason mentioned in 4.)
I think the setences are all good.
Since we can't have 'a fastest', there is something weird going on.
He runs fastest. He runs the fastest. 'fast' is an adjective, with the comparative forms 'faster' 'fastest'. Used as an adverb, it still compares. So there must be something in the context with which a comparison is being made. If you were talking about wild flowers, you would probably not suddenly say, 'He runs fastest.'
There is no 'a furniture' but there is 'the furniture', however, I can't imagine what 'a fastest' or 'the fastest' could be as a count or uncountable noun. I would analyze 'the' in 'the fastest' as an adverb of degree, an intensifier, the same usage as in 'the more the merrier'.
2. He runs the fastest of all the boys in his class. I think this sentence without 'the' is ok. Otherwise, see above. 'of all the boys' 'in his class' are just adverbials, so now you have four adverbs.
3. He is the fastest runner. Now you have 'fastest' as an adjective. 'runner' is a singular count noun. English normally requires an article before such nouns. 'He' and 'fastest' indicate that there is only one person involved, so forget 'a' and use 'the'.
4. She is happiest when she is alone. I think comparatives always compare. Here her degree of happiness is compared to her degree of happiness 'when she is not alone'. Again, this is not something you would just say out of the blue. There would be context to justify the statement. I think you may use the adverb 'the' if you wish.
5. This lake is deepest here. 'deepest' compares. It compares to parts of the lake not 'here'. It is deeper 'here' than elsewhere.