One other thing to add to your great response. REHEARSAL!
Orchestras rehearse as little as they do because conductors do everything you've mentioned. But additionally they facilitate a singular musical goal, and take the "democracy" out of it. Further to this is the time management. The musicians in the orchestra have enough notes to play and sound to think of, without having to consider how much rehearsal time we need/want to spend on a specific movement, how we would like to spend more time on movement 1 but actually movement 3 is notoriously tricky so we need to have time for it etc. Also, someone has to be monitoring the process to know when enough is enough, it's not always the case that more rehearsal time = better.
Depends completely on the context. In an ideal world the pianist leads the tempo.
However, sometimes the composition makes this near impossible (i.e. there is a large section of the orchestra that has to start playing in the new tempo, such that a conductor couldn't both react to the tempo of the soloist and show it clearly to the orchestra). If it's impossible for the conductor to get the tempo from the soloist, then usually there is time before the first rehearsal to discussing those spots, and discuss ideal tempos for the soloist. As you can guess, there is usually the possibility of negotiation, either verbal during this meeting, or during performance rehearsal both finding a more ideal tempo for the hall, orchestra, and days sensibilities.
Schumann piano concerto is a decent example. Second movement tempo is totally decided by the pianist, no discussion is necessary prior to rehearsal (unless they want some particularly abnormal). The second theme of the second movement is with solo piano in accompaniment position. So while we want to accommodate their musical needs, ultimately the musical burden is on the orchestra (in this case the cellos), as a result the tempo will be mostly decided (contextually of course, it has to fit with the first theme) by the conductor. Third movement beginning in theory is lead into by the conductor, however after the downbeat the soloist can show their tempo and the conductor responds.
I am a professional conductor (at a professional level, you can find videos of me conducting on YouTube if you wish). Feel free to ask anything you might like.
Eye contact is vital, and doesn't have to be reciprocated back to conductor. Peripheral vision is used a lot for front desk string players, back desk strings follow front desk (it's a hierarchy). There is usually more reciprocation from solo wind and brass players, if only because of their location on stage and their periods of rest (it's nice to have the safety net of a conductor cuing your entry after 53 bars).