I want download multiple Google Earth images randomly at chosen locations within the study area. Each image had a spatial extent of 250m×250m which corresponds, or exceeds the size of a pixel from medium spatial resolution satellite sensors (e.g. MODIS or Landsat),how to do this in R, and how to save these images?
I would suggest reading the Terms of Service for Maps and Earth API.
Paragraph 10.1 (a) says
No access to APIs or Content except through the Service. You will not access the Maps API(s) or the Content except through the Service. For example, you must not access map tiles or imagery through interfaces or channels (including undocumented Google interfaces) other than the Maps API(s).
That means you cannot download and save the imagery on your side, it will be considered as violation of ToS.