How strong does a character need to be to set aside some amount of money on a weekly gamble?
A person of strong character would not gamble in the first place. They would know how to spend their money and how to use money to make more money but keep the vices away.
What needs to happen to make it happen?
Nothing. Its more of a inborn thing. You can develop a strong character but its rare to do so after a certain age.
Does the great loss solidify our character?
Possible to happen. A person may see the reality of their addictions and how bad they have become and decide to use that hatred to stop gambling.
What exactly does it mean to discipline yourself in gambling (in the narrow sense)?
Stopping to play more.
Should we be disciplined in gambling, does it mean that we are better at gambling, or that we put less money at stake with more enjoyment, less fear, adrenaline, etc. ?
Like I said, a disciplined person would not gamble at all. But in a technical sense keeping that aside, being disciplined in how much bankroll you put in, having a limit and occasional stopping from gambling is good enough.