- Add a pilot's-side door to the Cherokee line.

Since the current design is integral to the structural integrity of the
cockpit, adding a door would not be a trivial change. In fact, it
might not be possible at all, and retain the current type certificate.
Personally, I've never found myself wanting a door on that side.

Piper knows how to build them (ie-Tomahawk).
It will add weight, as the framing required will require more AL.
Structurally, I don't see that there would be much difference. The
structure sits atop the wing. Will not the load be carried by monocoque
structure?

- Build an O-540-powered Arrow

They have done neither, and have thus been getting their asses waxed
by Cirrus and Cessna.

Ah, but note that Cirrus uses a fixed gear design. Maybe there just
wasn't enough of a demand for a big engined Arrow.

TIO-360C - 210 hp - 379 lbs
TIO-540W1A5 - 235 hp - 400 lbs

The Arrow is heavy enough without adding more weight to the nose.

Additionally, they could have made simple changes (like flush-rivets
and wing filets) to the airframe that would have at least given the
appearance of keeping up. Again, they have done precisely nothing,
beyond adding glass panels and upgrading interiors.

Look at how/where the fillets join the wing and fuselage.
While a fillet at the leading edge might pose a problem with removing
the cowl, the trailing edge fillet would interfere with the flap
operation.
The last redesign of the Cherokee series wing was 30-years ago when they
changed from the Hershey bar to the taper wing.