Yeah, you need to make sure that the pixels that are supposed to be transparent still have the same colors as the surrounding ones that aren't, or color bleeding like in the last pic will happen. Also you need to make the borders of the texture translucent if you don't want sudden clipping (if you want a texture to wrap around (e.g. a wall) then by all means make the borders opaque though).
There isn't really any workaround for the jagged edges, that comes with bitmaps. Either make them a higher resolution, or resort to vector graphics (at least to cut the borders). Although remember that most of the time you won't have sprites that close to the camera anyway, so the texels won't look as big in practice.