You should talk to your band director. I'm one :) Ok, so yes, the horn is held with the right hand, and the valves are operated with the left hand. That is the opposite of the trumpet.
Occasionally, the horn will have a trigger for the thumb. This kind of horn is called a "double horn" (as opposed to the single horn).
With the horn, the right hand is placed in the bell, and used, in part, to help control pitch.
The horn mouthpiece is more conical, the trumpet is more cup shaped.
The trumpet is a cylindrical instrument (fundamentally speaking), while the horn is fundamentally a conical instrument, with much more tubing.
As a result of the extra tubing, and the conical shapes involved with the horn, the partials are a little more difficult to hit with precision. You will find that the first few weeks or months on the horn will present some challenges with placing the notes.
Also, be prepared for a little more "back pressure" when playing the horn. The bore of the mouthpiece is narrower, which means that less air is allowed into the instrument (all other variables remaining constant). As a result, you might have to exhale before inhaling again (you might already be used to this, but it will probably increase).
Yes, the horn is typically in F. The trumpet is often in Bb, but is also seen in A, C, D, etc. All of this is rather negligible, as you're going to be playing with a different hand anyway.
The transition is a challenge, but frankly worth it in my opinion. I think you will really enjoy the sound you will eventually produce with the horn.
If your director suggested it, it means that he sees your talent, and wants you there. I'm a director, and I wouldn't ask someone to make that switch if I didn't feel they were qualified. You should feel honored, regardless of what you decide.
I am a high school band director