Hello SoH - I have suffered 2 major episodes of severe depression and anxiety - mad a complete recovery from the first one but now have intermittent depression since my last episode in 2010, so I know the torment of depression and anxiety.
I'm glad your DH is now getting some help for his illness. Is he taking meds and if so do you think they are helping him, or is he having some kind of therapy. Loss is almost always the root cause of depression, and this is the case for your DH as you mention a sibling died. My depression started following the loss of a very dear friend.
I think you are absolutely right to try to stop him drinking (many people with mental illness do self medicate with alcohol but it doesn't really help because alcohol is a depressant as you probably know, and after an initial high, then the depression returns) also you can end up with 2 problems (mental illness and alcohol abuse) and yes he should eat healthily and take exercise whenever possible.
I have read posts like yours before, and it's usually depressed men who are seen to be emotionally withdrawing from their partners and causing the sort of distress you are experiencing. The thing is depression causes us to feel emotionally empty and so we don't feel able to "reach out" to anyone, even partners. SO I don't think it's so much that he is pushing you away, it is more that he is pulling himself away because he doesn't have the emotional energy to "give" as he is "running on empty" so to speak.
The issue of him questioning his feelings for you are very common in depressed people. The problem is that this is the depression "talking" because he is quite right, he certainly doesn't feel like he used to - depression makes us feel like a stranger is inhabiting our mind/brain and we are powerless to do anything about it. It's a very scary feeling.
I also understand that he feels a sense of relief when you aren't there, but that isn't because of you, it's because he knows that he doesn't have to put you through any more distress. I read a post recently where the depressed man had actually moved out of the family home and was renting a flat nearby and kept in daily contact with his wife but said he needed to be alone and his wife was feeling like you are.
My partner is very understanding and supportive but I feel a sense of relief when he is out for the day as I don't have to put him through having to see my miserable face and my crying and my attempts to get through the day. I am completely phobic about anyone other than him (and 2 close women friends) seeing me when I am having a bout of bad days, as I feel ashamed of how I feel - and yes I know that's silly as it's an illness, but that nonetheless is the emotion I feel. The other thing of course is that the stigma about mental health is alive and well and that doesn't help at all.
YOu mention he looks like he's "going through the motions" of hugging you and playing with your little girl and that's a pretty fair description to be honest because that's about all you can do when you are so down. Anxiety and depression go hand in hand and it's often not possible to know which is which. There are some good techniques for handling anxiety (breathing exercises and mindfulness, meditation etc) but he may have already tried those. Most anti-depressants are meant to help with anxiety too.
You seem to be doing really well in supporting your DH who has a depressive illness, but I think the main thing for you to understand that he is not rejecting you emotionally, it's just that he is empty (running on empty if you like) and has to use his reserve tank to go through the motions. Not sure if my metaphors work, but the emotions I feel when the bad days come are emptiness, flatness, worthlessness, shame and sadness for my DP who has to see me like this. When the good days come I am "me" again and the world is a good place. You could try asking him about his emotions, although most men are not very emotionally literate.
Is this your DH's first episode of depression. If so 4 out of 5 people will make a complete recovery in 4 to 6 months, for others it takes longer, especially if he has delayed getting help. Depression is a self-limiting illness and it will end (I am just unfortunate that I get fluctuations) but that's unusual. SO please try to believe that there will be brighter times ahead. IN the meantime, continue supporting him and rid yourself of this notion that he is rejecting you emotionally. I know that's how it feels - I saw something on FB the other day saying "Don't always believe what you think..........."