Yes - you quite often hear "Forecasting the weather"/"rain is forecast", but it does just mean to "predict" the weather forecast. So we use it as both a verb and a noun. Often forecast is a casual comment - "Is rain forecast for tomorrow?", whereas "the weather forecast" is a little more formal "The weather forecast said to expect rain tomorrow", but it varies. For casual conversation, I often don't use forecast at all - "What's the weather say for tomorrow?" for example - people will just fill in the forecast part!

Yes. "To forecast" is usually a bit more scientific, and the person who "forecasts" is usually using some data and information about current trends. "To predict" just means a strong idea about the future, and may not be very scientific.