1. When you see "Britain lost the Battle of Britain," what does this mean specifically -- the Royal Air Force was eliminated as a fighting force and Germans obtained air superiority over England? Do you mean somehow Germany forced a British surrender without Sea Lion succeeding? etc.
It would be rather impossible to do that. What it probably means is that the Germans force the British to pull back their fighters north of London to recuperate for a few weeks, thus giving the Germans a window of air superiority over the channel which was what the prerequisite for Sealion entailed. While Fighter Command would have responded to an invasion by surging back down, it would have taken it time to re-establish itself and the Germans only planned for a window of superiority of a week or so. Enough to get the troops across or so they believed.
How do you lose the Battle of Britain but win Sealion? If you mean the Germans achieve superiority but not supremacy, that is hard. Eventhen, the Germans wash ashore and get crushed?
Not really. Assuming less skilled commanders on the British side (such as Leigh Mallory), German air superiority over the Channel for the period that the Sealion plan called for was by itself eminently achievable. Sealion would have still failed because the Luftwaffe, contrary to German expectations, would have still been unable to prevent the RN from steaming into the channel and sinking the invasion fleet. They would have made the RN pay a price in doing so but that's all they would have done.
Hitler wasn't seen as invulnerable quite yet. This just might lead to a coup.
Sealion was cobbled together in the aftermath of the Battle of France, which basically totally fortified Hitler's reputation as invulnerable. So yeah, he was. And if losing an army of 300,000 men in a massive land battle, something which the Germans were supposed to be good at, that turned the tide of the war wasn't enough to trigger a successful coup attempt then I fail to see how losing a few tens of thousands (at worst) in a amphibious operation, something the Germans have no skill in, that would in the immediate term have not apparently impacted Germany's position one bit would.
Hitler may hesitate attacking the Soviets and instead pursue the Mediterranean Strategy to salvage his pride.
If the Germans were seriously considering a long war against the British following the Battle of France, they'd never have tried Sealion. Goering might still have tried to bring the British to their knees with a strategic air campaign (basically the OTL blitz).
The whole Sealion plan was based on a desire to end the war quickly, which plans for a long war in the Med contradict. If the Germans were willing to do the latter, they'd never have risked the former.