If SNC's strength has a passive strength increase (Song of Summer Plot) that lasts over more than one phase, does the ability still protect it from being killed or discarded after the plot phase? What if my opponent has attached The Long Winter to my house card?

Stannis's Northern Cavalry

No attachments. Deadly.
Stannis's Northern Cavalry cannot be killed or discarded from play during a phase in which its STR has been raised or lowe red.

A constant effect is not considered to raise or lower a character's STR every phase. It happens once, then keeps on going (kinda like the Energizer bunny).

When you check a character's STR, you go through the equation "base STR +/- all applicable modifiers." What's more, each time you check a character's STR, you go through said equation from scratch, rather than starting with the last result and putting in any new modifiers. Because of this method for determining a character's STR, a character's STR is considered to be raised or lowered each time a new STR modifier becomes applicable. So for the purposes of SNC, the phase in which it's STR is raised or lowered is the phase in which the modifier becomes applicable for the first time (i.e., is first used in figuring out a character's STR). So for something like Song of Summer or The Long Winter, SNC's effect only kicks in for the phase in which the plot is revealed or the attachment enters play.