Use the Vector-> Research Tools -> Select By Location
The Parameters are:
- Select Features in: "Your point file"
- That intersect Features in: "YOUR RECTANGLE"
Then once that selection is made Save the new selection as a shapefile by right clicking the point file in the table of contents -> Save As:
Make sure to check the Save Only Selected, this will make a new shapefile from the Selected points within the rectangle.
For the other parameters you can use the Select By Expression here is a turorial: Here as opposed to SELECT BY LOCATION
You can use PyQGIS to automate the process