Is Rakshasa immune to the booming blade cantrip (SCAG, p. 142)? Does it avoid also the extra damage?
A Rakshasa is immune to the booming blade cantrip, unless they wish to be affected by it
The relevant trait of the rakshasa is as follows (emphasis mine)
Limited Magic Immunity. The rakshasa can't be affected or detected by spells of 6th level or lower unless it wishes to be. It has advantage on saving throws against all other spells and magical effects. (Monster Manual, p. 257)
As booming blade is a cantrip, and cantrips are spells of sixth level or lower, the rakshasa can only be affected by the booming blade cantrip if it wishes to be, as indicated by the quote and emphasis above.
Yes. A Rakshasa's spell immunity includes cantrips
Rakshasas have the following trait:
Limited Magic Immunity. The rakshasa can't be affected or detected by spells of 6th level or lower unless it wishes to be. (Monster Manual, p. 257)
Cantrips are spell level zero, which is lower than 6.
Cantrips—simple but powerful spells that characters can cast almost by rote—are level 0. (PHB, p. 202)