If I sell stocks, will I be able to withdraw cash from the proceeds that same day or will I have to wait until the T+3 settlement date? Does it depend on the brokerage?
No, you cannot withdraw the money until settlement day. Some brokers will allow you to trade with unsettled funds, but you cannot withdraw it until it is settled.
Think about it, when you buy stock you have to pay for them by T+3, so if you sell you actually don't receive the funds until T+3.
Depends on your account. If you have a margin account, then you can "withdraw" the margin, and it will get paid off/settled on T+3. However if it's a cash account then you will most likely need to wait. Call your broker and ask, each broker has different rules.