Distributing your own papers in your thesis I published three papers during my PhD, I'm now writing my thesis. 

I have written three chapters, each based around each paper, but re-written so as not to fall foul of self-plagiarism rules. However, I've still used ALL of the figures from the publication in the chapters. 

My concern is not really with this aspect of my thesis, though please tell me if I've misunderstood. 

Finally, in the appendix, as reference, I have included the PDF files of my published papers, exactly as they are from the journal website. 

Am I allowed to do this?

If not, am I allowed to use my own copies of the paper (e.g. the versions submitted on the arXiv)?
I don't understand why you are repeating the same material twice in your thesis. Each of your three papers is re-cast in a separate chapter in your thesis, and then verbatim in an appendix. 

You most likely have assigned your original copyright in your papers to the journal that published them.

I think adding the papers as an appendix is unnecessary, and may fall foul of the journal's copyright. The interested reader should be pointed to the journal article relating to each of the three chapters. 

If you reproduce substantially the same figures as that which appears in the journal -- to whom you've presumably assigned the copyright in the figures -- then you will need to ask for a license to reproduce those figures.