Wa'alaykumsalaam Wa Rahmatullah Wa Barakaatouh.
In regards to the importance of the 6 shawwal fasting, it is a Sunnah and it's is worth as if fasted a whole year or alife-time. The prophet Muhammad (Peace be upon him) “Whoever fasts Ramadaan then follows it with six days of Shawwaal, it will be as if he fasted a lifetime.” Narrated by Muslim, 1164.
Yes, these fast can be fasted as from the 2nd Shawwaal, which is the second day after Eid-u-Fitr.
These fasts can be fasted in different days. not necessarily consecutive.
Allah knows best