Inversion – Correct Usage of Weren’t+Subject and Were+Subject+Not for Inverted Conditionals

The word as in your examples is a conjunction.

It is used as a conjunction to express similarity. You can think of it (this as) as like, where like can be used as a more informal version of as. You can write it in normal word order ("as her mother is" or "as I do"), but keep in mind that you could also find the inverted word order used in a very formal style.

Here are some related sub-entries from Practical English Usage by Michael Swan.

326 like and as: similarity, function

We can use like or as to say that things are similar. We can also use as to talk about function – the jobs that people or things do.

326.2 as (similarity): as I do
As is a conjunction. We use it before a clause, and before an expression beginning with a preposition.
　　　 as + clause
　　　 as + preposition phrase
　　　 Nobody knows her as I do.
　　　 We often drink tea with the meal, as they do in China.
　　　 [...]
326.4 inverted word order: as did all his family
In a very formal style, as is sometimes followed by auxiliary verb + subject (note the inverted word order – see 302).
　　　 She was a Catholic, as were most of her friends.
　　　 He believed, as did all his family, that the king was their supreme lord.

Here is the related part of the entry 302 mentioned above.

302 inversion (1): auxiliary verb before subject

302.4 after so, neither, nor
In 'short answers' and similar structures, these words are followed by auxiliary verb + subject.
　　　 I'm hungry. ~ So am I.
　　　 I don't like opera. ~ Neither/Nor do I.

302.5 after as, than and so
Inversion sometimes happens after as, than and so in a literary style.
　　　 She was very religious, as were most of her friends.
　　　 City dwellers have a higher death rate than do country people.
　　　 So ridiculous did she look that everybody burst out laughing.

I think as most of the sayings go back to the olden days, old English, other languages, or different dilects, some of them seem to be incorrect grammatically.

The subject saying isn't common in use, it was used especially by some famous politicians, John F. Kennedy being one of them.

I think the OP is right that we should use don't ask instead of "ask not" to make the saying modern in English, but if you change the words of the saying, it'll spoil its beauty.