Quick question, easy to answer:

If the Runner got his 6th Brain Damage on his Action Phase, would the Runner lose the game on his Discard Phase because he cannot discard down to -1?

This depends. Lets say the runner has a hand size of zero, due to brain damage, at the start of his action phase. Actions 1 and 2 he plays diesel and draws six cards. Action 3 the runner makes a run and suffers 1 brain damage. At that point he would discard a card, now the runner has 5 cards in his hand and a maximum hand size of -1. The runner loses at the end of his turn since his maximum hand size is less than zero. From page 20:

If the Runner takes more damage than the number of cards in
his grip, or if he has a maximum hand size of less than zero at
the end of his turn, then he is flatlined and the Corporation
wins the game.

 The runner would lose the game immediately once he took damage that was greater than the number of cards in his hand.  Supposing he had 6 cards when he took the 6 Brain damage, yes, he'd lose at the end of his turn.

Ironburn said:

Ah, a hand size less then zero at the end of the turn. Got it, thanks.

radiskull said:

This is not precise, as it uses the words 'immediately' and then 'at end of his turn'.

There are two times the Runner can flatline:
1) Any time the Runner takes any form of damage, if they are unable to discard a card in response to that damage.

2) At the end of the Runner's turn when they reconcile their current hand contents to their hand limit, if that limit is -1 due to 6 Brain Damage Tokens.

I'm describing the two scenarios by which the runner can lose.  They are different.  We are saying the same thing.

 So let me see if I understand two things. 

1) Even if the runner's maximum hand size is some number, he or she can use actions to draw cards above that number but must discard down to the maximum card size at the end of the turn.

2) If the runner has a maximum hand size of -1 and has actions left to make a run and then is successful to steal an agenda to make it to 7 points then he or she wins before being flatlined at the end of the turn and losing. 

Is that correct?

1) Yes

2) Getting 7 agenda points overrides everything and immediately ends the game. This is even in the FAQ on page 3.

If the Runner steals an agenda from Jinteki using the core set
identity, but has no cards in hand, who wins?
The Runner wins the game. Whenever a player has 7 or more
agenda points in his score area, the game immediately ends. The
game ending takes precedence over any conditional ability that
triggers when an agenda is stolen.

 Thank you. This is one terrific card game. I am glad I gave up Mtg for this game.