According to this page on the etymology of the word 「切ない」, the 「ない」 in it is not the 「ない」 that means "there is no ...", but rather an ending that turns the noun 「切」 into an adjective.
On the other hand, there are many adjectives ending with 「ない」 that does seem to mean "there is no...", e.g. 「そっけない」、「あどけない」.
Is there any connection between the 「ない」 in 「切ない」 and the 「ない」 that means "there is no ..."?
If the answer to 1 is no, are there a whole series of adjectives ending with 「ない」 that acts as an adjective ending? If so, what are the most commonly used among them? How about such words as 「すくない」, 「きたない」, etc.?
This is like the English prefix in-, which may or may not carry negative meanings. In- in words like invasive or inherent does not have negative meanings, whereas in- in words like indifferent or insecure clearly has negative meanings. What does in- do in inflammable? We have to remember each word and get used to it.
ない at the end of some adjectives clearly mean -less, non-, etc., and often there are antonyms that end with ある. 心ない means heartless and 心ある means hearty.
ない at the end of some adjectives do not mean 無い. ない can be just another common word ending of adjectives. 汚い and 少ない are examples of this.
In a few cases, adding ない at the end of an adjective does not change the meaning. See my previous answer.