Do I need to open the chest to make it count as a win

The doors are tied to your health. The door will only be open if you have full health. However, the temporary HP boosts (blue hearts) can serve as a equivalent of the missing red hearts.

For example: If you're missing one red heart but have one blue heart the door will open. But if you're missing two red hearts and only have one blue heart the door will still be locked.

Collecting items, coins, keys, etc is an important part of becoming powerful enough to finish the later levels. If you skip the levels, you're unlikely to be able to survive. The difficulty ramps up quickly in the later levels, and you can't use "We Need To Go Deeper" to clear the final level - you'll have to beat a tough boss, plus a bunch of rooms filled with harder versions of earlier bosses.

Most of the items upgrade your range, firing speed, damage, movement speed, or health, and many of them can make it possible to move over objects or avoid damage. If you don't have some of these items by the time you reach the lower levels of the game, you'll have fewer chances to acquire items when you reach bottom, and you'll be weaker overall.

I've been unlucky enough to find few "good" items on a run to some of the lower levels of the game, and the bosses down there are tough without the proper equipment.