I have been watching many videos of concert band performances which has a soloist standing just beside the conductor. Let's now assume that we are talking about a trombone concerto with a solo trombone player, should he/she be looking at the conductor while playing or listening to the music of the band and watching the audience?
The conductor should always follow the soloist, not the other way around. The conductor is there to keep the large ensemble of musicians playing together smoothly. The soloist is there to take liberties and add some personal interpretation to his or her part. If it's a concerto, for instance, the conductor's job is to follow the soloist and wave his arms to keep the ensemble in sync with the soloist. If there's a section where the soloist is not playing, then the conductor is in charge and can add his or her own interpretive touches.
So the conductor and the soloist must be aware of each other at all times, but in certain sections the conductor leads, and in certain sections the conductor simply follows the soloist and uses conducting gestures to communicate what the soloist is doing to the rest of the ensemble -- who may not be able to see or hear the soloist in concert, for that matter.
The soloist should do all of those things. You don't necessarily have to look straight at the conductor though. Peripheral vision can be enough to see the movement of the hands and baton. If you just listen instead of watching too, there's a good chance you'll miss something and not be together with the ensemble.
However, the conductor also needs to watch the soloist for phrasing and tempos. Some soloists are more demanding of this than others, and some conductors are loath to relinquish control to the soloist. There's a famous incident between Leonard Bernstein and Glenn Gould which I'll let you read about here. http://en.wikipedia.org/wiki/New_York_Philharmonic_concert_of_April_6,_1962
The answer to this question depends on the standard of the accompanying group.
If the soloist is playing with an amateur/community group they will need to watch the conductor because the conductor can't rely on an amateur group always following the beat.
If it is a professional group the conductor will follow the soloist and the soloist does not need to watch the conductor. In a professional group the conductor can rely on the group following the beat.
They should watch, or at least be in contact with, each other. The conductor has to lead an orchestral entrance, but he does it at the mutually agreed moment. An ability to have this sort of rapport is one of the job specifications for a conductor. There's a story about management trainees being taken to watch a string quartet work and perform together. Whether it is true or apocryphal, they would have learnt a lot about several strong personalities arguing, deciding and then fully co-operating in the service of the music.