I made a similar suggestion a while back on uservoice, but it was declined at the time. I'll reproduce it here since it is similar (but not exactly the same) as your suggestion.

Tag suggestions tooltip steals focus when dismissed

1. Start creating a new question.
2. In the "Tags" box, type "java".
   => The suggestions tooltip appears, showing "java", "javascript", etc.
3. Click in the main edit area (where you would actually type the question).

Expected: Focus goes to the question area so I can type my question.

Actual: Focus goes to the question area very briefly, then the the tag suggestions tooltip disappears, and when it disappears it gives focus to the tags edit box.

I made a similar suggestion a while back on uservoice, but it was declined at the time. I'll reproduce it here since it is similar to (but not exactly the same as) your suggestion.

Tag suggestions tooltip steals focus when dismissed

1. Start creating a new question.
2. In the "Tags" box, type "java".
   => The suggestions tooltip appears, showing "java", "javascript", etc.
3. Click in the main edit area (where you would actually type the question).

Expected: Focus goes to the question area so I can type my question.

Actual: Focus goes to the question area very briefly, then the the tag suggestions tooltip disappears, and when it disappears it gives focus to the tags edit box.