The hadith says that everything is idle and play except for four: caressing one's wife, archery, learning to swim and taming a horse; is this correct, and is this authentic? And will learning how to shoot a gun suffice for archery given that he said archery and swimming?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

‘Uqbah ibn ‘Aamir reported that the Prophet, sallallaahu ‘alayhi wa sallam, said, "Everything with which the son of Adam amuses himself is vain except for three (things): shooting his bow and arrow, training his horse, and playing with his wife." [Ahmad, Ibn Maajah, At-Tirmithi graded it hasan (good)] The hadeeth does not mention swimming.

Shooting with modern weapons is as meritorious as shooting with bows and arrows. The Prophet, sallallaahu ‘alayhi wa sallam, said, "Verily, strength is in shooting." [Muslim] It is one form of the "power" that Muslims are enjoined to prepare for the purpose of jihaad against the disbelievers. Allah, The Exalted, says (what means): {And prepare against them whatever you are able of power...} [Quran 8:60]

Again, we draw your attention to the fact that the hadeeth which you asked about does not mention swimming. However, swimming is one of the sports that are commendable according to the Sharia. There are some reports encouraging Muslims to learn how to swim.

Allah knows best.