Other pets in the house can be greatly affected by the loss of a pack mate. If you can, you should let the other dogs of the house sniff the pack mate that passed away. This will let them know they are dead and they won’t be as likely to look for them everywhere.
If your dog is showing signs of depression, don’t love on him while he is in this state. Get treats or toys out to snap him out of it and when he shows the least bit of interest you reward that! Give him more attention then usual, like more exercise and playtime. Cuddling on the couch doesn’t count. It needs to be physical activity where the mind and body is stimulated.
Try changing things up a bit. Go to new parks and blocks when exercising your dog. That way there are no memories in that environment associated with the other dog. You can also do this in the house by getting new dog beds and toys and getting rid of the old ones. Change the routine more by walking differently, maybe jogging, new treats, and find playmates in the neighborhood.
As far as leaving your dog when you go to work, if he is now alone, it will be very different than he is used to. Try leaving for short periods of time like around the block. When you come back in see what his state of mind is. Is he depressed in the corner? Is he showing over excitement? Or is he normal?
If he is showing signs of depression when left alone, higher a dog walker to come and play and walk him in the middle of the day. If he is over excited it is because he isn’t getting any stimulation during the day anymore. You will need to spend more time with him exercising before you leave as well as when you get home. Also, you need a dog walker or at least a neighbor to visit with him for a while. Start working with him on more obedience to stimulate his mind more. Also, look into getting toys that role and drop treats out, or putting peanut butter in kongs for him to lick out. This will increase activity while you are gone.
Do not look into getting a new pooch until the family is emotionally stable again. For help on how to get over a beloved dog yourself click here.