Yes, it's time to remove the crib. You can either get him a toddler bed (maybe the crib mattress will fit it) or get him a twin bed with rails so that he won't fall out of bed.
It's not really important whether he sleeps in the bed or the floor - just so that he stays in his room. If you don't fight him on WHERE he sleeps in his room, he will end up sleeping where he is most comfortable. (And that's probably the bed...)
If the door alarm doesn't continue to keep him from wandering around at night, which is a safety concern, then put a lock on the outside of his door.
The less stimulus there is from you at night, the more chance that he will get bored and sleep. It will take time for him to self-soothe. Right now he's trying a new thing - getting out of the crib.
For naptime during the day, if he is taking 2 naps, drop one of them. If he is still not willing to nap during the day in a bed, then he is probably dropping his daily nap. That's hard on him and you, but toddlers sometimes drop their naps.