I am a victim of physical abuse and forty years later I'm still going to therapy about it, partly to make sure that I don't act the same way towards my son the same way my abuser did.
I remember my abuser (my nanny) smacking me with her shoe because I didn't eat my sandwich. She told me if I told my parents what she did they would know then what a bad person I was. I believed her. Your 3YO might be encountering the same behavior from your wife.
When I got sick, she told me that I got sick on purpose just to make her life miserable. One time I was so afraid of her that I said I was OK even though I felt awful and I ended up with a 103 degree fever.
She used needles on me when I didn't look awake enough for breakfast. Needles were better. Bruises left marks. She also ripped up my stuffed animals on occasion just to make a point. She also gathered dead bees and would threaten to press the stingers against me.
But you know what, that wasn't the worst part. It was the constant, daily walking on eggshells that wore down. It was never knowing when I walked through the door who I was going to get, Dr. Jekyll or Ms. Hyde. It was knowing that the neighbors heard her screaming down the block and NOT ONE PERSON told my parents what was going on. When I got to high school, I signed up for every fricking after-school activity I could think of to get out of the house. I ended up with having several bouts of depression, with a couple suicidal ideation bouts built in there just for kicks. I told people of my fantasies of slitting my wrists and letting the blood seep into the hallway. My college required me to call the infirmary on a regular basis to assure them I was still alive. And I was LUCKY. My sister, who was subjected to even worse abuse, ended up turning to weed and alcohol.
I would rather die than subject my son to the same childhood. I would never ever want my son to go through what I went through. You have NO IDEA what you are setting up your children for if you don't leave that house. Your wife was arrested for a DUI in 2011 and yet you still have a three year old? I'll give you the benefit of the doubt - perhaps she had a period of sobriety then. But you're with her now. And you have every intention of staying.
You talk about health insurance and bills as the reason that you stay with your wife. Let's forget the fact that you're letting a 3YO live with an alcoholic and are essentially teaching her that being drunk is normal. You are exposing your children to so much financial liability it makes my head spin. Just one lawsuit against your wife (if you're still married to her) could wipe you out.
Yes, a firm financial foundation IS a very good idea when you're raising children. But when your house is burning, do you say "But we put so much money down on the down payment? We can't leave!" No, you get out!
I wish you strength and fortitude in the days ahead. You're going to need it.