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It would be a huge improvement to the accessibility and semantics if the p tags were changed to label tags with for attributes.

Yes, I pretty much did not read past the <p>Name:</p> point.

I opt for a P around both the label and the field, gives greater CSS hooks and encapsulates each form area nicely.

That's what I generally do as well...

Fact: you can wrap input elements in a label element. No extra for attributes required, either.

Yes but if you want to float the label and have it cleared in the next "row", that's where the separation of elements comes in handy.