No they do not automatically become a US Citizen. They would have to get a spouse's visa and then a marriage based Green Card. After the elapse of time - usually 5 years, they may then apply for US...
An Indian cannot become a US citizen by marrying a US citizen. Not only Indians, but any other person from another country cannot become a US citizen just by marrying a US citizen. A US citizen can...
Yes, but they have to apply for and receive US immigration permission .
Answer.
Do not work until you have your Employment Authorization Card. You can get one by filing the I-765, which you should...