The Melachot of Shabbat

A couple of weeks ago, we saw a mouse in the house and put out a cage trap. A mouse was trapped last Shabbat. Because we felt bad for the mouse, we took it (on Shabbat) to an isolated area and let it out. Did we violate anything by trapping the mouse on Shabbat, or by letting it out (like some melachot that come in pairs, like tying and untying)? Was the cage muktzeh and, if yes, did tza’ar ba’alei chayim justify taking it out?

A couple of weeks ago, we put out a cage mouse trap. A mouse was trapped last Shabbat. Are we aloud to set it out During Shabbat?

Is it permitted to remove loose hair on Shabbat, which usually includes ripping it, that has gotten stuck in a necklace?