When you are sight-reading, how far ahead in the music do you look?
This is highly individual and depends on the complexity of the music. Taking myself as an example, when reading very basic music, I can read, say, four measures at a time. For most music I can only read one or two at a time, and for complex music, it could be as little as a few notes.
On the other hand, I've encountered highly talented sight-readers who even with complex music can read as much as two complete lines at a time.
However, with experience, the amount one looks ahead can increase.
Do you somehow look at the treble clef and bass clef at the same time?
Yes. This is essential. When learning a piece for oneself, reading hands separately is fine, but for really playing music at sight, reading both staves is necessary.
The typical advise, which I follow, is to read from bottom to top.
After you've done it for long enough you don't have to think about it anymore.
This is true, at least up to a point. The more one reads, the more one begins to recognize certain patterns and musical structures. Scales, for example, are pretty easy to recognize (first, that the notes make a scale; later, what the actual scale is). Root position triads in close position are also fairly easily recognized. Other things, like recognizing a broken chord or some other pattern of notes, comes as you go.
Sometimes, though, thought can be required. Large leaps, unfamiliar structures, multiple voices ... these can give one pause. Experienced readers learn to ignore what they can't immediately interpret so that they can keep going through the music. The ignored parts can be addressed later on.
I also assume you can never look at your hands while sight reading.
Certainly the less often the better, but when learning to read it can also be helpful — it allows a visual connection between the notes on the page and the keys on the keyboard. Also, of course, for something like a large leap, a look might be needed.
Sight readers do sometimes look at their hands, but it's typically a very quick glance, just to quickly locate the note(s) which they can then move to by muscle memory. The keyboard might also be within one's peripheral vision.
For more ...
There are lots of sight-reading-related questions here, so searching for the piano and sight-reading tags will provide a wealth of tips. My own strategy/philosophy for improving sight reading, which is how I've found the most improvement in my own skill is at
Acquiring advanced level sight reading
Although the question related to advanced reading, the tips apply regardless of level.