Is Facebook illegally using your device processor?

You are giving access of your whole phone data when you have installed their app. They can do anything if they want to, because you have already given them the permission.

No they are not Illegally using it Facebook does not go onto your Device unless it came with the Phone you got or you choose to Install It 

Unclear what you mean by "illegally using 

your device processor". Do you mean like 

running programs in the background that 

I did not authorize to be on my device? 

Something like bitcoin mining, or ...?

i would hope not

I just found six pictures on Facebook that came off of the SD card on my cell phone I did not post them there.

Nobody else has those pictures except the SD card on my telephone the only way they could have got there is if the Facebook app posted those pictures.

I did not caption those pictures on Facebook nor did I write the little story. 

Apparently Facebook has some kind of ability to take whatever they want off of your cell phone if you install their app.

No.  Don't have the app installed.