Sorry for the reason you've come here Dreaming. From what you say I think relapse is the wrong word. You have to stop drinking to relapse and yes, he is taking top-up nips during the day. Of course he's lying - have a look at the 'Quackers' thread to see many similar examples, even in the face of clear evidence to the contrary.
He drank when he was left alone with your 6 year old. You've had to stop him picking up the kids because he's incapable of controlling himself. I get it - I'm a sober A myself, but it's adding to your burdens, not helping.
At the moment he only wants to stop drinking in theory, but not enough to go to AA, or work some sort of program. Worse, he denies drinking when he's about to drive his daughter. If you hadn't had the breathalyser would he have driven regardless? Yes, because protecting his drinking is his first priority.
Q1. He drank during the day - you just didn't see him.
Q2. If he's going to spend the money on drink, then you probably will have to buy the groceries etc. Think about ordering online and getting it delivered.
Q3. He's not relapsing because he's not stopped drinking. Expect it to be continuous from now on.
I know you're hanging on to hope now, but it doesn't seem he's motivated at all. Many people only get it after the marriage has broken up. Right now he has no work, a comfortable home, almost nothing to do. It's not helping his self-esteem or depression, and it's providing zero incentive to stop drinking.
Sorry this is not a hopeful answer, but you may have to take control of a lot more family tasks, and assume he's not going to be much use until he starts taking recovery seriously. You may be able to organise help for yourself with child-minding and and house cleaning. Perhaps he could get a job, anything, to help pay for it.