Basic question on calculating real GDP
First, note that GDP is not the quantities of all goods and services produced in that year multiplied by their prices in some base year, but the quantities of all final goods and services produced in that year multiplied by their prices. Intermediate goods and services are not included.

1) GDP is about value created in the economy. So, if there are new services or products, we should simply include them. For the price, we take the price in that year and deflate it using a factor that accounts for inflation. Note also that new services and products typically come at the expense of other products produced. People that would otherwise be, say, working at a factory, are now programming apps.

2) As you would with oranges and apples. Lawyers typically charge a price per hour. So, it would be number of hours times the price. For more general services, like, say, the police force, you would just calculate the costs of the police force for a given year.