The MCU data sheet will be of use if you want to get really low level with the Due. However if you have the Arduino runtime in there, then using the libraries that are available is possibly a simpler way to get started.
If you are going to use the ESP8266 for the WiFi interface, why not have a single image frame saved in that which is accessible from the web server software it will contain.
That way the Due can interface to the camera, receive an image, pass it on to the ESP8266 and forget about it. The web server in the ESP8266 can simply replace the old image with the new one (you probably will double buffer to prevent flickering) and then pass that on to whoever might be connected when they refresh the web page.
As for the number of SPI interfaces, the DUE has a hardware interface but also can have software SPI interfaces. Don't forget that an SPI master can (if the design and access requirements are right) talk to more than one slave (just not simultaneously) using the appropriate \SS\ line (which should always be used to provide synchronisation anyway).
SPI may not be the best way to communicate between the Due and the ESP8266 - the UART might be better as you really only need 1-way communication. ALong the same lines, if you have not yet bought the LCD display then you might investigate one with an I2C interface (could be a bit slower but the Due has both hardware SPI and I2C interfaces).
I get the impression that you don't have a great deal of experience with microcontrollers and the like. While your goal is certainly achievable, I would suggest that you start with something less ambitious and work your way up. Even just displaying a fixed image on the LCD graphic display can be a challenge; once that is working then add the camera, then the WiFI module and finally the SD card (be aware that accessing an SD card can be slow, especially compared to the speed of the images coming in from the camera and being processed on the display and WiFi - you will need to work out how to handle the various processing speeds of the different functions).
Susan