First off, I'd have to ask, do you love him? I know he's screwed up, but do you love him? I've been in your shoes. I didn't realize how I was screwed up for choosing to be with someone with an addiction problem. I joined Alanon and also a group based on the book, "Women Who Love Too Much." To this day, nothing I've ever read, no seminar I've ever attended has changed my life like that book. I've bought dozens of copies and given them away.
While the two of you are separated...for now...just please know that its a roller-coaster. You may get back together, or not. You may hate him the first year, then want him back. Don't be surprised. Ask me how I know this.
Is there some way you can gather friends and family together to stage an intervention?
Will you check to see if your health insurance will cover a rehab program?
Three year old's can only process so much info. Try to keep it simple, as in, "daddy's going to have one house and I'm going to have one house. So, you'll have two homes and daddy and I will each have a bed for you, toothbrush for you, etc. at each place." Baby steps. Don't overwhelm. Try short visits with dad that can be extended to overnights. Try to keep the emotional stuff out of the explanation for now (except the 'mommy and daddy love you soooo much'). Try to just stick with the details that will matter to HER right now, i.e., your special dolly can go with you to mom's AND dad's. You'll have some clothes at mom's and dad's.
Be prepared to buy two of everything and have duplicates at each house. Also, be prepared for one shoe here, one there. The soccer shorts are in the hamper at one house, and the rest of the uniform is in the other parent's car. Ask me how I know this. Its aggravating. Take proactive steps and just be the one who's extra-prepared. I even packed extra snacks in the kids special "bag" (a Land's End enormous canvas boat bag with our last name embroidered on it), which became the go-back-and-forth bag. I was worried about whether my ex would remember to feed a young one snacks...because he's the type who 'didn't want to' get it about stuff... Need I say more...
Best of luck to you during this difficult time. It will get better over time...
C.