Yes, the broker, exactly! That's even the trick to increase the leverage! Just study this page on the maths: https://en.wikipedia.org/wiki/Margin_(finance) As a self-clearing broker one can define the margin requirements as one wishes, up to the minimum possible by law or exchange rules etc... The less the margin requirement, the higher the leverage.
Sure, but as i understand it the broker is the one giving you the margin since they have excess cash on hands. If you are your own broker/clearing firm with only your own money deposited, how are you going to give yourself margin? If i had a brokerage account with $10K deposited and a 1:4 margin have $40K buying power. If i open a position for $40K, the extra $30K comes from the broker as a loan, since they have excess cash they can do so. If you start your own broker clearing firm with $10K in capital you will never be able to open a position beyond the $10K in cash since there is no excess capital. I could be wrong but i think you believe you can set whatever margin you want, but you still need the buying power to open the position. A margin of 1:4 doesn't mean you only have to pay 25% of the stock price, you still pay 100% of the stock price but 75% of the money (or shares) required is a loan from your broker. Who is going to give you that loan if you are your own broker/clearing firm?
You just can open more positions! This is maybe not the right example, and unfortunately I don't have the right example at hand, but in the past I once analyzed this and did the math and came to the conclusion that one better should be a broker oneself to increase the leverage. Sorry it escaped me how the maths was. I know it works, but don't have the algo at hand right now. I think this works primarily with options, futures, and currencies (fx). Check the examples here: https://support.kraken.com/hc/en-us/articles/203053116-How-leverage-works-for-margin-trading
There is no logic in this at all. Leverage isn't created out of thin air. If there is no one to give you a loan there will be no leverage at all.
Proceeds from short sales or options writing could be used to provide leverage, at least mechanically. I'm not sure what is allowed by the regulations for very large firms.
You only need to have the margin requirement. With this, you can control a multiple of that as shown in the example in the link above, ie. at https://support.kraken.com/hc/en-us/articles/203053116-How-leverage-works-for-margin-trading And as a broker you can self-define the margin requirement, up to regulatory minimums. Just study how the leverage factor of Lehman Brothers was... :
Lehman Brothers borrowed money to invest, they a debt-to-equity ratio of 30 to even 60 at times. So for each $1 they had in capital they had up to $60 in debt at times. Again, who is going to give you that loan?
As i have said again and again, when trading for a broker with 4:1 margin, you only need to have the margin requirement since the broker is providing the rest of the capital as a loan ... If you have 1000$, 4:1 margin and you purchase $4000 of stock, you put up your own $1000 deposit and you are $3000 in debt to your broker. You provide $1000, the broker provides $3000. You provide 25%, the broker 75%. Who is going to give you that loan if you are your own broker ...