No, I can’t think of any cases. „ You chased me to my room“ would be the must natural. „Away“ usually indicates that someone chased you to a non-specific place. For example; You chased me away.

Yes, because you chased me away. Where? to my room

Depending on context: Someone literally ran after you: You chased me to my room. Someone scolded you until you went to your room: You chased me away so I went to my room. The way that you have it is a sentence but it doesn't sound quite right for some reason, having a bit more context would help.

If you were to say this, no one would question it. You can absolutely put “to” after “away.” ex. I ran away to the bus station. This makes perfect sense.