>Technically, any gene sequence can be achieved with enough time and resources.
not in a meaningful way, no. the probability that a new mutation you want will occur is much much lower than the probability you can breed offspring without a gene that's already in the bloodline.
Once a desirable sequence modification is identified through artificial means, what is often done in practice is to simply expose samples of the organism to UV until the desired sequence appears "naturally." The output of this process is not typically considered GMO, at least for regulatory purposes.
not in a meaningful way, no. the probability that a new mutation you want will occur is much much lower than the probability you can breed offspring without a gene that's already in the bloodline.