What a sad thread this is, with some very ignorant view points.
As said, depression (and mental illness) affects people very differently and ranges in severity so no one here can say exactly how the OP's brother is affected without knowing him and sitting down to talk to him over a period of time. The term depression can be a misued as a broad term for a range of mental health issues. Who knows what thoughts or feelings are going through the brother's head.
It can make people selfish, deeply sad, scared, overwhelmed, mean, desperate, joyless, self loathing, suicidal, irrational, destructive, unpredictable, hopeless, helpless, angry, implusive, anxious, paranoid, insecure, exhausted, insomniacs, and any other other 'undesirable' and 'unnapealing' and 'unhelpful' trait that you can think of.
The depression (and thoughts/ feelings) may completely mask any other good traits they have (or did have) like being kind and loving, funny, skilled, helpful, generous, active, confident etc etc. None of us excep the OP can say what the brother was like before he was ill.
Some people with depression would rather live and die under a bridge or in the their bed than live with life, themsleves, claim benefits or be with other people, so ill are they.
That is the trouble with mental illness - it makes you mentally ill. And the more ill you are, the less you can see reason and what others would consider good sense. If you can work and take proper care of yourself and others despite some difficuties with your mental health, then you are (thankfully) actually just less unwell than some other mentally unwell people.
So whilst taking on constructive activites (and this might not be paid work right now) will help his recovery, the fact that he has depression may mean that he is simply unable to recognise that this is a good idea and be unable to make himself do it.
I'm suspect his home instability and financial problems are both a source of and a symptom of his illness. Situations like this can be a vicious circle. He may be cycling from deep depression to desperation and an urge to (ill-conceived) action back to deep dperession again. This happens with depression not just Bi-polar. Even depressed people can get urges of motivation (or claustrohobia/panic/ desperation or just genuine healthy motivation), but can't always sustain it long enough to make the decision work out. There is no argument that this is really awful to deal with. Perhaps it is the wife that is deperate to find something that will cheer him up?
It won't have helped that they were loaned/given money for a car but then had to pay it back. Ths could have thrown them both and it can be very hard to make the best decison when under pressure.
Perhaps they thought that getting puppies would be constructive becasue it would give him something to care for (this can encourage taking care of oneself) and help him exercise. Not all bad, but sadly not through through financially.
There is so much here that is an unknown.
No one on here should be passing judgement on whether or not he is genuinely ill or is faking based on a few snippets from the OP. He may not even feel able to tell her the whole truth of how he feels. No doubt there are some really messed up and desperately awful situations going on with the OP's family but slagging off people with diagnosed mental illness is not the answer.
If he is unable to work then he needs to (or be helped to) reapply/ appeal his ESA decisions and maybe apply for PIP if he is having trouble caring for himself. There are organisations such as Mind and Rethink that may be able to help. Also the CAB. It isn't easy or pleasant to go through the process, but would help them financially. Being found fit to work previously does not mean anything.