
Not sure about the corgi in the ring but the dachshund should be untrimmed although I suspect may be 'tided'. As this is a cross I think your common sense as a groom would tell you what is needed but I would have though a good comb through to the skin and then a bath would be all that is required for a happy health dog apart from perhaps a trim of the nails and any over grown feathering.
OH dear, don't think this is a cross but two dogs, in which case Dachs has a soft straight or nearly straight coat with feathers and the Corgi a double coat may be dense particularity if neutered and not well cared for.