You wouldn't want to capture the space key because you'll need that in search queries but the arrow keys sound like an idea for those interested in accommodating both cases.

You pretty much never want a space at the start of your search query though, so you could DTRT based on whether there's preceding text there or not.

Good point but I think it's now coming to a point where confusion might come into play. It's a balance between inconveniencing someone with auto focus and confusing someone else.

Arrow keys and page down sound like "activate autocomplete" to me.

Before you type something? That seems odd.