Is it usual to write referee reports in the first person?
Resoundingly, yes.  At least in my field (number theory).  I just looked at the 15 most recent referee reports I received, and only 1 out of 15 of these reports avoided writing in the first person.
Note: I restricted to reports on initial submissions to avoid obvious duplication of referee styles.  Of the 15 I looked at, the reports were written in sufficiently different styles that I'm confident in saying there wasn't too much duplication of referees in this sample.
Added: To address the &quot;mishmash&quot; aspect, in none of these reports did the reviewer refer to themself in both the 1st and 3rd person, but I have seen it before and have possibly done it myself.  The bottom line is that it doesn't really matter.  Referee reports are not formal pieces of writing meant for public display (like parts of grant applications or MathSciNet reviews), and while some reviewers may prefer to refer to themself in the 3rd person (presumably to make the review sound more formal/professional or less personal), I think most prefer to write more directly.