Okay. Not going to design a room and spec out everything for you.
Just some advice. If you have 3000 for a display I would look into a large Plasma or LCD. Most if not all new displays have an HDMI port. As long as it has a component in you can use the display as your switcher. Control systems (Crestron, AMX) are expensive and expensive to program. A projector will need lamp replacement ( $200 - 400US) about every 2000+/- hours. You can squeeze more use out of them but the picture quality will degrade. LCD/ LED/ or Plasma should last 120,000 hours. I have not seen a projector last close to that regardless of lamp replacement. The optical drive usually goes and then it is around a $1200 to 2000 repair depending on model. You should be able to get a nice audio system for 3K no problem. Stay away from Bose.
Just some advice. If you have 3000 for a display I would look into a large Plasma or LCD. Most if not all new displays have an HDMI port. As long as it has a component in you can use the display as your switcher. Control systems (Crestron, AMX) are expensive and expensive to program. A projector will need lamp replacement ( $200 - 400US) about every 2000+/- hours. You can squeeze more use out of them but the picture quality will degrade. LCD/ LED/ or Plasma should last 120,000 hours. I have not seen a projector last close to that regardless of lamp replacement. The optical drive usually goes and then it is around a $1200 to 2000 repair depending on model. You should be able to get a nice audio system for 3K no problem. Stay away from Bose.