Is it possible to cast banishment or dismissal on a druid when he is wild-shaped into an elemental?
Not with the intent to send them to an elemental plane.
Wild Shape, as it pertains to elementals, is the equivalent of Elemental Body. In this spell, you assume the form of an elemental, you do not fully become one. You remain planar, as your origin is still in the default plane, regardless of the form you assume.
Banishment only works on extraplanar creatures and it sends them to their Home Plane. So if your Druid happens to be extraplanar like say an Assimar, then yes, but not because they are in elemental form, only because they are in fact extraplanar. If the druid is from the material plane, first they are not extraplanar and second they are already in their home plane. Spell descriptors: "It enables you to force extraplanar creatures out of your home plane." "Will save or be banished back to its home plane."
No. The wild shape ability allows the druid to assume the form and certain abilities of an elemental but not the crucial bit needed for banishing. The druid's creature type remains the same - he/she does not become an outsider with the extraplanar subtype. For something to be banished (with dismissal or banishment for instance), the creature must have the extraplanar subtype.