How to use "the ...est of the ..."?

No, none of these sentences would be used. The lamest of the lame is slang, and is an insult, one we don't use very often. In other words, it's not a phrase you can apply to all adjectives. People will know what you mean if you say it, but it's not typical English.

You are talking about adjectives (words that describe something).

There is the basic adjective, for example: pretty, tall, dark.

Then there are the 'comparative adjectives' where something is compared with something else. It may, for example, be prettier, taller or darker. These are all comparative adjectives.

Then there are the superlative adjectives where something is compared with everything else or all of defined sample and is better than all of them. The superlative adjectives often end in 'est', for example: prettiest, tallest, darkest.

The 'est' suffix is not used with all adjectives.  In your example of the word 'beautiful' the comparative would 'more beautiful' and the superlative 'most beautiful'.

It is not particularly meaningful to say 'the cutest of the cute' or 'the most beautiful of the beautiful'.  This construction is used for effect rather than to add to the meaning.  If someone is 'the most beautiful' they MUST be more beautiful than other beauties.  If they are 'the cutest', they must be cuter than everyone else.  To call someone 'the silliest of the silly' adds impact to their degree of silliness but it is not a construction that would be used often. It IS grammatically correct but a sentence can be grammatically correct as well as being unusual or even meaningless.

Best of the best. Or the worst of the worst.