The butterfly has a very rich symbolic history. In contemporary culture they (arguably) stand for freedom, chance and change, but classically they refer to the psyche.
In Paprika, butterflies seem to appear as a first sign of a transition between reality and dream, and to signify an invasion into (or rupture from) the subconscious. But it happens only - I think - when the subconsciousness is invaded directly, without the use of a dream, as this is one of the features of the DC Mini that is told ought to be constrained throughout the film. It is also why this only happens
when Dr. Seijirō Inui, a.k.a. the Chairman, pervades the unknowing minds of Himoro, Shima, Tokita, and later even that of Konakawa and Chiba/Paprika.
The only exception to this is when Dr. Tokita joins the dream of Himoro.
This can be explained by him using a freshly constructed DC Mini: after all, as far as I know, the allergic reaction (the anaphylaxis✝ they talk about) spreads through the collective (un)conscious by means of infected DC Mini's.
Another explanation, since the butterflies seem to show up exclusively around Dr. Chiba, is that they represent lucidity. This ties in nicely with the following observation:
When Paprika is caught by Osanai, she is - like the specimens on display in the wooden cases on the walls around her - pinned to the table like a butterfly:
The symbolism of the scene in which Osanai rips her from the appearance of Paprika, is illustrative of the significance of the butterfly in the film: ripped as from a cocoon, Chiba drops to the floor, devolved to a pupa-like stage, nude and wingless - she loses her lucidity, as she 'sheds' Paprika:
This scene, evoking classical mythology, reminds of Psyche, the ancient Greek character from the story of Psyche and Eros, whose name has become synonymous with 'soul' or 'spirit' (or, naturally, 'psyche'): in the visual arts, she is often depicted with butterfly wings. Wings are often associated with the mind, like Hypnos ('sleep') and Morpheus ('dream') (since Ovid, apparently).
I don't really know where you get the idea that "a dog sitting next to a kennel appears before and after dreams in Paprika". As far as I know, the only time it is cut into the film in a suspicious way, is when Dr. Shima jumps out of the window, and we see this dog looking up:
Although his action is induced by a dream, we see Shima's jump taking place from reality, and the dog is cut in right before he jumps, probably because Shima was throwing a frisbee with him a short while ago. It seems to be there only to build suspension, although it could refer to the good-heartedness and faithfulness of the doctor.
✝ Additional observation: paprika (or bell pepper) is a common cause for anaphylaxis.