What's the difference between "as", "when" and "while"?
The doorbell rang as/when/while Anna was asleep.
Which is right and why?
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While is used only about a continuous state, and another event or state that happens during that time. It does not imply or refute causality.
The doorbell rang while I was making dinner. - single
I listened to the radio while i was making dinner. Continuous
When implies a causal relationship between two things: when X happens Y happens. It can be used about a single event, an intermittent state or a continuous state
Please come and see me when you are free. - single
When the red light is showing, you can't cross the road. - intermittent (whenever)
When we were young, life was simpler. - continuous
When as is used about time, it implies two events or states happening by chance at the same time
I saw her as I was leaving. - event
The doorbell rang as she slept. - event/state
The sun was setting as the boat sailed away. - state/state
As @Peter pointed out, as can also mean because.
The phone rang as she was sleeping. (time)
She didn't answer the phone, as she was sleeping. (because)
The best word to use in your example is while. When is not suitable because there is no causality. As is possible and clear in this case, but may be ambiguous in similar situations.
In your examples
The door bell rang when Anna was asleep.
The door bell range while Anna was asleep.
both are understandable as the door bell rang during the time Anna was in a state of sleeping.
A better second sentence might be
The door bell rang while Anna slept.
A bit more problematic is using as
The door bell range as Anna was asleep.
As can be used to either show causality or simultaneity.
The door bell range as Anna fell asleep.
would mean the bell rang just as she started to drift off to slumber land
As Anna was asleep, the door bell rang.
because Anna was asleep, the door bell rang
would mean Anna's being asleep caused the bell to ring, which would be odd, but possible if it was one of those nights where she was interrupted by people constantly ringing the door bell at the most inopportune times.
No, "As Anna was asleep, the door bell rang." In this case her being asleep is the reason for the doorbell being rung. It's not people constantly ringing the bell, as then her being asleep would not have made a difference.
Let's say Anna wanted to see the Perseids, but wasn't sure she could stay awake long enough. Her neighbour offered to check every hour if someone was still moving in her apartment, and to ring the bell if everything was quiet. This went well for the first few hours into the night, with Anna waving at the neighbour every time he checked. But then the neighbour couldn't see her anymore. As Anna was asleep, the door bell rang.
OK, not quite: To get the bell as a seemingly independent entity, you'd need an independent observer. Maybe we're seeing this from Bella's point of view, who lives in the same appartment and has been dozing all this time. She's vaguely aware of the proceedings, including the point where Anna was too tired and had decided to take a nap. Now the neighbour was trying to raise all the star-gazers. but ss Anna was asleep, the door bell rang.
Something like that, at any rate.
The correct one would be while Anna was sleeping. It is not when because sleeping is an ongoing action, not a completed action (a word with -ing at the end in this situation would be while because it is still happening). When refers to a time that a single action happened, but while refers to things that are occurring at the same time. As only works if it is the first word in the sentence for a single, instantaneous action. (Example: "As Anna woke up, the doorbell rang.")