Is this ethical?
I don't see this as a question of ethics. Is it unethical to write a bad or not useful paper? No, it's simply the best that some people are capable of (or interested in) achieving. By not including implementation details, the paper's authors are clearly limiting the usefulness and future impact of their papers. Nonetheless, the paper may still have some usefulness, so I don't see why anyone would think publishing such a paper is unethical. The only exception is if the authors are dishonest about their claims of what the algorithm can do or how well it works, and are deliberately hiding implementation details to prevent their dishonesty from being exposed. Obviously that would be unethical.
How to proceed when many things presented in the code are not
explained in the original manuscript?
Do your best to fill in the holes. If it seems too difficult or too much work to reconstruct the missing implementation details yourself, talk to your professor -- he or she may not realize that they've given you an unreasonably difficult assignment, and it would be their job to figure out how you should proceed.
What if the code disappear from author's page? Will the results exist
in the universe once again?
Again, it's up to the author to decide how much work they want to invest in ensuring their paper has a long life and makes a meaningful contribution. It's also up to the journal editors and reviewers to enforce some minimal standards. But at the end of the day, if the author doesn't care enough about their work making an impact to deposit their code in a repository that would outlast their personal web page, that's really their decision, but it doesn't reflect well on them, and potentially limits the cumulative usefulness of their work to the community.
To summarize, what we are seeing here is an example of short-termism, which is something you see in all walks of life. Some authors write a paper just with the goal of getting the short-term reward of getting their paper published (and the attendant professional rewards the academic environment will give them for such a publication: jobs, promotions etc.) and will just put in the minimum amount of work to achieve that immediate goal. Others care much more about the long-term impact their work will have on the research community, which will also eventually translate to a personal benefit to them since they will develop a reputation as better researchers. It is the researchers who belong to the latter group whom you usually hear about as the famous, super-successful authorities in the field whom everyone admires, so I certainly recommend trying to follow that approach yourself.