Paramhamsa is actually the name of a class of ascetics.
Other classes are Kuticaka, Bahudaka, Hamsa etc. A Paramhamsa is considered as the one who have spiritually evolved the most among these classes.
Several references to them can be found in the minor Upanishads and other scriptures.
Here is one such reference from the Narada Parivrajaka Upanishad (...
Yes Lord Muruga is a Guru of Lord Shiva and to explaians the meaning of OM mantra to him. I'm not sure about whether it is mention in Skanda Purana or not but it is mentioned in Kantha Guru Kavasam. In that there is a line " Siva Guru nadha" which means he Lord Muruga is guru for Lord Shiva.
Image Courtesy : Dheivegam
Om is a Mantra, so chanting Om constitutes Mantra Japa. Other names for it are Pranava Mantra, TAraka Mantra etc.
Om represents the Adi Dvani (primordial sound) and is an Ekakshara Bija Mantra (mono-syllabled seed Mantra). It also represents the Brahman in the MAntrik level.
EkAkshare tathA kute traipure mantra nAyike | Shtri datte swapna
lavdhe cha ...