If you plan to have this available in Reader, you need to start the process of obtaining a Reader Integration Key. Details in the SDK. Understand that Adobe reserves the right to refuse you that key and so you may not be able to develop for Reader (or not with 100% equivalent functionality)
How you contact your own app is up to you. There are numerous IPC methods that you can chose from.
Sure, you can obtain the text of the PDF. And you can do whatever you want with it.
Yes, you can generate a raster image of each page of the PDF in whatever size and resolution you choose.
Plugins to Acrobat & Reader are DLLs written in C/C++. Details are in the SDK.