To begin with, "Lucky Numbers" are a sequence of numbers generated by a sieve similar to the Sieve of Eratosthenes for finding primes.
It starts with the set of natural numbers. Begin by selecting the second element (which is two). Now delete every second element (with respect to order). Now move on to the next element in the set. Take this element to be $n$ (which would have been untouched by the previous iteration). Delete every $n^\text{th}$ element in the set. This continues as $n\to\infty$ and what you have left are the "Lucky Primes".
Now because this is recreational mathematics it is usually not taken seriously. They are generated with computers and observed. My question differs.
Is there a rigorous way to approach the "Lucky Numbers"? Such that they can be studied in more detail?
Can properties about them be determined without the sieve? Or are they as "random" as the prime numbers?
Are there similar constructions used in non-recreational mathematics? (Except for the primes).
Finally. Is there anything mildly interesting about them currently? Such as an open question etc.?
There are indeed open questions on lucky numbers (see your link), e.g., whether or not there are infinitely many lucky prime numbers, or infinitely many twin lucky numbers. Also the analogue of the Goldbach conjecture is open. It seems to me that one may approach lucky numbers by sieving theory and other tools from prime number theory. The similarity to the behaviour of primes is certainly interesting. The "recreational" aspect is perhaps only mildly interesting - if you are lucky.