Yes you can do trading in US market through authorised Indian banks for pay in and pay out , though the remittance of amount per year cannot be more then  USD 2,00,000 as mandated by RBI guidelines.

Many full-service Indian brokers (like ICICI direct, HDFC Sec, Kotak Sec, Reliance money etc) has a tie-up with the foreign brokers. They have made it very simple to open your overseas trading account with their partner (foreign) brokers.

That completely depends on the stock exchange of US where you wish to do margin trading

Also the settlement will be done in cash, so the currency which is permitted by the exchange will have to be used

1) You can do forex trading and that also in Indian currency.

You can do so in India as well you need to speak to ICICI Bank who is providing this marginal trading in New York stock exchange from your Indian Bank account

Margin trading also refers to intraday trading in India and various stock brokers provide this service. ... A margin account provides you the resources to buy more quantities of a stock than you can afford at any point of time. For this purpose, the broker would lend the money to buy shares and keep them as collateral.

To facilitate you to do the same, an Indian stock broker enters into a tie-up with a foreign broking partner who has the license to act as an intermediary and execute the trades on your behalf in the foreign markets.
The Indian stock broker will act as an introducing intermediary between you and the foreign broking house. The Indian stock broker will also help you in getting your account opened and completing the formalities of Know Your Customer (KYC) applicable for that country.

As per the remittance norms of the Reserve Bank of India (RBI), an Indian citizen can remit a maximum of USD 2,00,000 in a financial year, from any of the authorised banks in India, including for investments in international capital markets.

It takes around 24 to 48 hours to remit money from your bank account to your trading account with the foreign broker and around 48 to 72 hours from your trading account to your bank account.
Once your account is opened and funds are transferred, you will be provided a client Login ID and password to have an immediate access to the foreign brokers trading platform to buy and sell shares of the listed foreign companies. All dealings like trading, delivery of shares/funds etc. will be done directly with the foreign broker without any involvement of the Indian stock broker.

Any trading in which your proceeds are received after deduction of tax is legal.