Did DB Cooper's aircraft land safely with the aft stairs open?
The aft airstair was partially deployed for the duration of the approach and landing.
The transcripts of ATC communications between NW305 and various ground stations are freely available (and a fascinating read). The relevant communications (with Reno) begin on PDF page 87 (numbered 213). Below are the mentions of the airstairs from that point on, with times given in PST, 1971-11-24. I added the text in brackets for clarity.

22:35, NW305: OKAY AH WE'RE TRYING TO MAKE CONTACT WITH [Cooper in] THE BACK [of the airplane] NOW AND AH WE'RE GOING TO GET THESE
STEPS UP BEFORE WE CAN MAKE OUR LANDING...
22:43, RENO: NORTHWEST THREE ZERO FIVE WHAT'S THE STATUS ON THE STAIRS NOW SIR
22:43, NW305: AH WE HAVEN'T BEEN ABLE TO GET AHOLD OF ANYBODY [in the back of the plane] YET UH TRYING TO CONTACT HIM AND THEY'RE STILL
DOWN SO I WE HAVEN'T DECIDED YET AH WE MIGHT COME IN AND LAND WITH
THEM DOWN THERE'D BE SOME SPARKS SO
22:48, NW305: APPROACH AH THREE ZERO FIVE WE CAN'T SEEM TO RAISE HIM BACK THERE WE KIND OF HATE TO WE'D JUST AS SOON LAND WITH THE
THING HANGING DOWN IT ISN'T ALL THE WAY DOWN AND THEN PROBABLY WON'T
HAVE ANY PRESSURE ON IT SO WE HOPE IT'LL (unintelligible) FREE...
22:57, NW305: ...WE'RE NOW MARKER INBOUND AND UH AH WE'RE GOING TO BE LANDING WITH THE AIR STAIR IN THE DOWN POSITION AH WE DON'T SEEM TO
GET ANY RESPONSE FROM OUR FRIEND IN THE BACK
[The aircraft touches down and is taxiing.]
23:03, NW305: SEE ANY SPARKS COMING OFF THE TAIL AT ANY TIME ON AH TOUCHDOWN
23:03, RENO: NEGATIVE NONE AT ALL THE ONLY THING THAT'S VISIBLE ON THE TAIL IS LIGHT AH APPARENTLY ON YOUR RAMP
23:03, NW305: ROGER UNDERSTAND STAIR LAMPS AH STAIR LIGHTS
23:03, RENO: I DO SEE SOME AH SPARKS NOW JUST A FEW AH TRAILING YOU AH AS YOU'RE TAXIING IN
23:03, NW305: OUR AH PASSENGER TRIED TO DISEMBARK
23:03, RENO: I STILL DON'T AH GETTING A FEW SPARKS NOT VERY MANY AT ALL
23:03, NW305: OKAY WE'LL TRY TO SLOW OUR TAXIING HERE AS MUCH AH IT PROBABLY IS DRAGGIN AH

Why did the crew not retract the airstair before landing?
The airstair can only be operated from two positions: one in the main cabin and one on the exterior of the aircraft. It cannot be operated from the flight deck. The testimony of Tina Mucklow, the flight attendant whom Cooper interacted with almost exclusively, reveals that

Approximately four minutes after takeoff, [Cooper] stood up, told [Mucklow] to go to the cockpit and close the first-class curtains, and for no one to come out from behind the curtains.

It should be noted that the airstair was still closed and locked at this point. In fact, it was Cooper himself who performed the deployment:

...he asked her to demonstrate to him the procedure for opening the rear door and extending the stairway. She did this and was under the impression that he understood how to do it. She commented that it occured to her at this time that this was the only function of the aircraft which she had discussed with the hijacker during the time she had been with him which he did not seem to be fully aware of.

Even though before Mucklow left Cooper in the cabin &quot;...she pleaded with him to take the bomb with him and he said he would take it with him or disarm it...&quot;, nobody could be sure, so complying with his demand for no one to enter the cabin was the obvious choice. Even after the crew suspected he had jumped, there was still some chance that he was present. That—on top of the clearly dangerous situation of being unrestrained in an airborne jet transport with a door open—compelled the crew to sit tight in the cockpit and land with whatever configuration they could control from there. Indeed, third officer and right-seater on the flight Bill Rataczak stated that

...no member of the crew departed from the cockpit to check on the presence of the hijacker following the arrival in the compartment of hostess Mucklow, who had locked the door behind her.

How did the airstair remain partially deployed on its own?
The airstair is in contact with the tarmac when deployed on the ground. In fact, it's designed to essentially wedge itself against the ground and act as a tail stanchion to keep the airplane from tipping backward during loading and unloading. Early 727s such as N467US had mechanical locks for the airstair in both the extended and retracted positions to accomplish this function.
However, the crew had evidence that the airstair was not locked, as Mucklow &quot;observed the red indicator light go on the [flight engineer's] panel indicating that the stairs had been lowered.&quot; On the 727-100, that light would be de-energized only if the airstair was locked either up or down. (On the -200, a green light was added to indicate a fully extended and locked position.) The ATC transcript indicates the captain's later belief that the airstair wasn't &quot;all the way down&quot;, so it's reasonable to assume the warning light remained illuminated.
That the airstair didn't lock down when deployed indicates that it never fully extended. It is likely that the air loads simply prevented full extension. The airstair hydraulic actuators were not designed with the strength necessary to overcome a 170-knot airstream, especially on early 727s. The extension was limited by the airspeed and subject to fluctuations with changes in the airstream and the weight of Cooper upon the stair.
The inability to fully extend the airstair in flight was even known on the ground beforehand, as the transcript of communications between NW305 and the airline's base at MSP reveal (I cleaned these up):

18:38, MSP: DON'T KNOW OF ANY WAY TO LOCK STAIRS IN AN INTERMEDIATE POSITION.
18:40, NW305: ...STAIRS WILL OPEN ABOUT 20 DEGREES AT 120 KNOTS. IS THAT ENOUGH FOR AN INDIVIDUAL TO ESCAPE THE AIRCRAFT?
18:40, MSP: WITH HIM ON STAIRS WILL OPEN POSSIBLY ENOUGH FOR HIM TO GET OUT. REMAIN UNPRESSURIZED. BE IN LANDING CONFIGURATION WITH
FLAPS. SLOW TO [APPROACH] SPEED BEFORE TRYING TO EXTEND STAIRS.

As it turns out, the stair could not be opened much if at all at 170 knots during a climb, and the captain had to &quot;have 'er about down to a hundred and sixty knots&quot; and level off at 7,000 feet to get the resistance low enough. It makes sense that a higher angle of attack would cause a greater air load on the door due to the increased frontal area. It also makes sense that the airflow alone would be insufficient at any speed to completely re-close the door due to the geometry of the empennage, the weight of the stair, and some hydraulic damping. Once the door was open, the crew were able to climb again to 10,000 feet and keep the airspeed &quot;in the vicinity of one seventy, one eighty [knots]&quot;.
Although the hydraulic actuators would have provided damping, the swirling wakes from the landing gear and flaps—not to mention the aircraft turbulence inherent when flying low and slow over mountains in a storm—undoubtedly wrenched the stairs up and down considerably. Cooper's exit has been romanticized as a high-dive into the unknown, but it was more like trying to exit a springboard during an earthquake, with his own weight just exacerbating the problem. We know as much from Richard McCoy, Jr., a copycat hijacker who survived his jump (at the same airspeed) and was later captured and interviewed:

...[McCoy] stood geared up in the open doorway for a very long time before
descending the aft stairs. It was almost midnight, the sky was black,
and the wind was freezing, yet his hands were sweating. Finally, he
was able to collect his wits, take a deep breath and step outside onto
the stairs.
As he stepped out, the entire stairway dropped two feet under his
weight. Knocked off balance with his 70-pound duffle bag between his
legs, he grabbed the right handrail and wrapped both arms around it,
desperately holding on. The stairs shook and vibrated. The noise from
the jet engines on either side of him was horrendous.

In fact, it was cabin-pressure fluctuations caused by Cooper descending and exiting the airstair that allowed for the determination of exactly when he jumped (8:11pm PST):

Crew reported oscillation of cabin rate-of-climb indicator. This probably due to [Cooper's] weight now on stairs, stairs extended further, resultant effect on cabin pressure.

The FBI's analysis was that &quot;When [Cooper] jumped, the air stairs were forced closed (by outside air pressure) causing an increase in air pressure similar to shutting a car door with the windows up&quot;. Although we've established that the door was not forced completely closed and locked, it's clear that the airstair was able to hang somewhat freely in the airstream, its deflection affected by air loads, the weight of Cooper, its own weight, and any dampening provided by the actuators.
What was the risk of landing with the airstair deployed?
The crew knew from the cockpit indication that the airstair was neither down and locked nor up and locked. But they had prepared somewhat for the idea of landing with a deployed airstair before they even departed Seattle:

19:27, MSP: IT WOULD PREVENT ANY SUBSEQUENT TAKEOFF IF YOU LAND WITH [THE AIRSTAIR] DOWN.

Mucklow advised Cooper more than once of this fact, given that Cooper had stated intentions to get to Mexico and that Reno was just a necessary refueling stop along the way due to Cooper's wanting to be flown low and slow.
Landing with a deployed airstair could have a range of consequences that had never been investigated or analyzed, so most likely the airline was playing it safe. At the very least, a detailed inspection would need to be carried out before the airplane flew again. If the stair were downlocked, the results of a landing would depend on what failed first. If only the latch failed, you might get away with just dragging the stair, which would cause some exterior damage but overall that'd be a relatively minor fix. If nothing on the airstair failed, I suppose you could bend the entire fuselage. I'd also worry about bits and pieces of failed components being rocketed around the empennage or into the cabin.
Fortunately, at this point the crew was safe from the worst outcomes because the stair was not downlocked. In fact, as we've seen, the crew knew that even at 120 knots the airstair would not be deployed far enough to strike the ground upon touchdown. That speed is in the ballpark for an empty 727 on landing, but even at a lower speed it's entirely plausible that the airstair did not strike the ground on touchdown, and was only able to fully extend once the aircraft was at taxi speed.
What damage did the aircraft sustain?
Given everything we've covered so far, the damage was probably not extensive. Dragging the stairs along at taxi speed would mean not much more than some sheet-metal repair. Looks like some of the railing paneling sheared away, probably in the air:

There's also the placard found in the forest a few years later that got ripped away at some point during the flight.
In any case, the crew was able to easily deplane via the deployed aft stairs:

23:14, NW305: ...WE'LL GO DOWN THE BACK STEPS AND AH AH WE'LL BE TAKING LEAVE OF THE AICRAFT AND SHUTTING IT DOWN AT THIS TIME...