Yes, just re-publish for changes to go live.

Will the original survey link still work or will you need to use the new one?

The survey link(s) remain the same.

Yes, once republished the future respondents will see the latest version 

Will the responses gathered in the first version be saved? Or they will be annulled?

@Mateusz_Rewerelli All responses of all versions will be available in the data and analysis tab.

However, if you delete a question, then you won’t see that when you download the responses.

If you add a question, then the responses of the first version won’t have answers to that.

If you modify stuff, then the new question/choises will appear in the download tab.

To see the old version and answers, restore that particular version in the survey editor.

The best practice is to create an ED called version and update it before every publish. This will ensure that you know which responses correspond to which version.