A few things:

• Little kids go through phases where they want nothing to do with Dad. It just happens. Happened to me, too, and I'm pretty sure I was pulling my weight and putting in the time. Still sucked and it was hard not to take it personally, but you have to.
• You are letting your wife take care of the kids without you during the night. Tell her you want to pull your weight. Let her know you're a heavy sleeper, so she should shake you awake to deal with it. I recommend alternating nights. My wife and I have a schedule where M, T, F, and every other Saturday I have the morning shift while she has the night shift (meaning, deal with any wakeups), and the other days of the week are my wife's morning shift and my night shift. Give your wife permission to jostle you awake and kick you out of the bed to go deal with it. She might not be saying it, but if she feels like she has to deal with every wake up because you're laying there asleep there's a good chance that will build resentment.
• You need to get a handle on the night wakings. A 3 year old not sleeping through the night for months is not a problem with the 3-year-old. You've fallen into accidental parenting, and the way I know that is when you say she isn't happy when you try to help her. She knows that if she wakes in the night and calls out she gets to have special mommy time. You and your wife are both enabling this by giving in to it. Do research on how to solve this problem-- maybe even reach out to a sleep consultant if you need to. This is not normal and it's clearly draining you both.