First, I will point you to FINRA's page Day-Trading Margin Requirements: Know the Rules. It covers all of your questions and more.
To answer each of your questions:
A day trade is when you buy and sell the same stock on the same day. It doesn't matter what order, or if you dosell short and then buy on the trading insame day. If you execute a buy order andopen a sell orderposition for Stock A inand then close at least part of that position on the same day, that is a day trade.
It needs to be for the same stock. If you buy shares of stock A and sell shares of stock B in the same day, that is not considered a day trade.
Related question: Will this trading activity flag my account as PDT (Pattern Day Trader)?
The day trading minimum equity requirement comes from the Federal regulations. Once your account is flagged as a pattern day trading account, you will be required to maintain $25,000 in equity in the account. This amount can be in cash or in "eligible securities."
In your example, if you held $20,000 in cash and $30,000 in stocks and mutual funds, you would meet this requirement.
Related question: Pattern Day Trade - $25,000 Margin Account Rule
You cannot combine the equity in different brokerage accounts to meet the requirements; each account that is flagged as a pattern day trading account must meet the equity requirement in order to be allowed to trade.
From FINRA's page:
Can I cross-guarantee my accounts to meet the minimum equity requirement?
No, you can't use a cross-guarantee to meet any of the day-trading margin requirements. Each day-trading account is required to meet the minimum equity requirement independently, using only the financial resources available in the account.