2. The House will never elect a non-member as Speaker, regardless of the legalities.
3. Dole is very old news. His day is past. Plus, even if point #2 is ignored, proposing Dole for the position would likely lead to a revolt among some of the more conservative members of the GOP caucus.
4. Open White House meddling in the selection of the House majority leader/Speaker would cause great unhappiness in the House.