I cannot answer your question on Halifax.

Fighter Command was on its knees when Goering switched to bombing British cities in retaliation for Bomber Command's raid on Berlin

I disagree with this assertion. The Battle of Britain was a battle of attrition between the RAF and the Luftwaffe, and it was one the RAF was winning. You can find most of my rationale here. The British consistently overestimated the strength of the Luftwaffe while the Germans consistently underestimated the strength of the RAF. Also, the RAF was able to replace its losses in fighters and pilots during the Battle of Britain while the Germans were unable to do so. The perception was that the RAF was on its knees and they were hard pressed, but the reality was that the RAF was winning the Battle of Britain throughout its duration.

Was this the critical mistake its made out to be or would the RAF have won anyways?

The above link gives my answer to this question as well.

Strictly speaking, your first part is a what-if, so not really within the realm of /r/AskHistorians to answer, but I'll give it a shot and say no. The posts below, and much prior reading on the Battle of Britain suggests that the Luftwaffe was never close to victory, nor did it necessarily have the forces to even ensure this. A defeat leading to a 'deal' is highly unlikely.

Moreover, Halifax was a man with poor health, this being the grounds on which he turned down the PM-ship in favour of Churchill, and was a loyal Briton. It is unlikely that he would have either become PM in the (extremely unlikely) event of a British defeat in the BoB, and even if he did I doubt he would be willing to reach a deal, considering that German hegemony of the continent was a direct threat to the Empire Halifax would be trying to preserve.

TL;DR: Hitler: Deal or no deal, Herr Halifax? Halifax: NO DEAL!