Is this ceiling box wiring correct and how can I reconnect an abandoned 3-way switch to it?
That part on the left can't be right. If you closed the sink light switch it would trip the breaker. That is a dead short from hot to neutral.

The black tape is not a mistake. That should be the feed to the switch and the black is the switch leg that returns to the light. Then the light is connected to the neutral with the others.

The black wire to the 2 gang box should be used for the hot or the red but not both. Right now you have the black wire seriesed through the light.

I don't think they make a duplex switch with a three way in it.

Does the dining area ceiling box feed a fan/light? You have a hot feed going to it along with a switch leg. 

You are attempting a pretty complicated setup for your first time. There is a little bit of a learning curve to this.

You may need to post a more extended diagram of the devices and the wires running between each. 

Sorry I can't help more.

Edit I see your extended diagram explains the kitchen light switch was not a dead short but just power to the switch. You had the white marked black but have corrected that in your last drawing. Using red for the switched power and travelers is the better method as others pointed out. Your final drawing looks electrically sound. With your current wiring it is not possible, but the NEC requires a neutral at switch locations with a few exceptions. If you are not being inspected you could consider this 'grandfathered' wiring. In the future, you should take a neutral to the switch location.