Yet I cannot proof my residence as I have been most of that time (sadly) on the streets

Currently thanks to an inheritance and a lot of luck things have gotten much better

I only have my Belgian passport currently (and driver's license). I have nothing else as official papers. My bank account is in Belgium.
Can I apply for ILR if I now rent a room in London or Cambridge? Can I just after a year of ILR apply for citizenship?
Will the procedure be any easier for me as I am a EU citizen and therefore can stay as long as I wish in the UK even without ILR/citizenship?
How to deal with the lack of proof of residence/official papers? Should I just fill out the ILR application with this limited information?