You can (and should) take the nestbox out as soon as the eyes are open. They might be small in size - but once the eyes are open they can do all the same things as a full grown rabbit: eat solid food, drink water, explore, socialize, etc.
In my experience, the sooner the nestbox is removed, the faster the kits hit butcher weight. While in the nestbox, their nutrient intake is limited to the once or twice daily nursings from mom. Out of the nestbox, they still nurse from mom, but they also eat solid food which results in far more calories than nursing alone and gets them growing fast.
There's no need to lay it on its side. The kits will hover together and stay warm with or without a nestbox.
Hope this helps.