Can I submit my old PhD thesis to arxiv with the date set to its original submission date?
I don't think so.

The "proper date" for an eprint is the date you upload it to the arxiv.  If the arxiv allowed people to back date their work (by up to ten years!), then it would at best confuse and at worst totally ruin the use of the arxiv to sort out chronology and claims of priority.  So while I don't know that they don't allow this, I have never seen an example, and I hope that they don't.

What you can do, of course, is to include in the comments that this is your thesis from 10 years ago.  If you do that then everything seems clear.  If you are worried that your thesis appearing on the feed/list of new papers will cause it to get more attention....wait, isn't that why you are posting it to the arxiv in the first place?