As with Mrs P your opening concerned me. I understand what you're saying but you come across as unable to relinquish control to others, accept that they may have well-founded opinions that differ from yours and/or to feel that your post will not be adequately self-explanatory - perhaps because you know or fear that you are missing something out? Were you to talk to me in that way we would not begin a conversation. Nothing irks me more than being told what I think or how I am going to react. If you want my opinion you have not only to ask but to listen, actively, seeking to understand, not while constructing your rebuttal in your head.
Ok, so that aside, I agree with others that you need to talk and a counsellor might help. A few other thoughts:
Your comment about money rings alarm bells. Do you spend frivolously while your wife feels obliged to be frugal? Is that because you regard your earnings as yours not the family's and don't have equal 'spending money' or equal say over luxuries?
Do you always seek resolution to arguments, perhaps too quickly, before your wife has had a chance to calm down? Does she feel that you are being bossy or seeking to take control of every issue? Could you come to some agreement that any argument needs to be resolved before bed time and that sulking beyond that time is unacceptable? Some such parameters might give her the opportunity to calm down and instigate discussion herself.
Sulking is immensely irksome and childish and often manipulative. People can storm off because they are too angry to be coherent but you both need to be able to come back later, state the problem, discuss the desired solution and sort things out if you are to have any chance.
One thought, inevitably, is that your wife has given up on the relationship or actually wants to break up but feels tied to domestic circumstances and is just dragging things out, unpleasantly. She need to be challenged, or challenge herself to consider, arrive at and state a view on that.
Your wife sounds very unhappy. She may feel helpless. She needs to be able to identify what she wants and how she is going to achieve it. This might mean doing quite small things that will make a difference but does involve her taking some responsibility for identifying a course of action. It may be better put by your asking what you can do to help e.g. can you offer / arrange childcare while she goes out to exercise, see friends, or some other activity?
Your comment about a romantic weekend does read like a kneejerk response I'm afraid - proposing an instant solution rather than seeking to understand the problem. It suggests a failure to understand that your wife is unhappy and detached from you and needs to address more basic issues in order to regain a normal equilibrium before she could possibly feel the necessary enthusiasm and motivation for a celebratory event. I'd expect someone with experience of depression to be able to relate to that. People do this a lot though, usually to avoid having to confront others' unhappinness (like thinking that someone grieving, recovering from a break-up, or depressed, will be cheered up by a 'night out' when what they really want is comfort and conversation from close friends).
I can understand the idea that because of an accumlation of small problems in the past, it is no longer possible to believe someone when they claim they will change. If a problem and its solution have been pointed out and explained but continue to happen repeatedly, then after some time, why would anyone believe change is likely or even possible? Your wife may well feel that the examples from the past are relevant when she feels let down in a way that follows a theme. You need to find out what that theme is. Alternatively she could be saying you should feel lucky to have her and how dare you criticise her. Incidentally, do you 'look up to' her, respect her intellect, values and capabilities?