So I have now graduated to operation sunflower, and I had a couple of quick question. Can I use "give us the tools" as the British during the first strategic phase to set up an extra two boxes on the map? I assume so but as there is no deployment phase maybe I'm wrong.
You are right, you cannot put them on the map.
And if the British gets eliminated on the very first player turn, do I still get to deploy my strategic reserves? The rules say that you only check map control status at the end of turn housekeeping phase but I thought I read that you yield control the moment all your units are eliminated.
Yes you can deploy. The check you are talkign about happens only at the end of a GAME TURN (not player-turn). Check rule 15.0, second paragraph, "End of Turn Segments".
And finally, how come the map is uncontested at the start as there are units from both sides on it?
This is a Scenario exception to represent Rommel's surprise attack. See is as the Game-Turn started a bit before this: The Axis entered the map, and the Allied units did nothing in their own turn; the Axis were not "really" present on the map when the scenario started.