My understanding is second stage is harder, first because of the rocket equation. Every Kg on the second stage is much more costly than an additional Kg on the first stage.

Second, the second stage needs to go into orbit before it can return. That's much more speed and a harder reentry.

Third, you have to deal with thrust. Falcon 9 has 9 engines, which means the landing burn can uses 1 engine for 1/9 of the thrust. This is still too much power and F9 currently needs to do a suicide burn. You can't do that with only one engine on a stage that is 1/10 lighter than the first one.

Fourth, second stage has a vacuum engine with a nozzle not fit for atmospheric use. It's so thin and flexible it would probably crumble or disintegrate upon entry.

You could parachute it assuming it survives reentry, which is the first hard problem. The second problem is that chutes have to be replaced and are very tricky to install to work reliably. Plus they're another component. Second stage being lighter helps with parachuting it down. Also have to calculate if the extra weight of the chute makes it economical still.

Hmm... well I doubt parachutes are any harder than landing a rocket on a barge, so I'm sure this could be done. Sounds like one possible path. The stage may sustain some damage on landing but it would allow at least some of its more costly parts (engines, electronics) to be re-used and the rest of the material to be recycled.

It's a lot more difficult. Every gram you add in reuse facilitating systems is a gram lost in payload, so every gram has to be incredibly valuable. That's a daunting constraint to deal with.

Consider the worst case scenario here. You have a launch to GTO where the upper stage goes out to geostationary orbit altitude over the course of several hours before returning to LEO. The stage needs to be able to have enough power generation to live that long (currently it doesn't), which probably means solar panels or bigger batteries. The re-entry burn will take a little propellant, which isn't a big deal but does reduce payload capacity again. The re-entry will require new avionics and thermal protection systems. Then landing will require new control systems to steer toward the landing site plus new engines and possibly new propellant storage for the landing. At this point the main engine can't be used because it not only has the wrong thrust, it won't even work at low altitude.

But that's not the worst part. The worst part here is that being in orbit the stage is now "floating" free from the Earth's surface, which means that it now has landing windows in the same way that there are launch windows into orbit. For a high altitude orbit like a geostationary transfer trajectory this is very problematic because you have to wait for things to line up, but you only get roughly two chances a day so you might be waiting a long time. Lowering the apogee would help (you'd probably be able to achieve a landing within a day) but that's costly in terms of propulsion.

The obvious easy way out of that mess is to move the problem boundaries by going to 3 stages. The 2nd stage would only push to LEO and re-enter after a once around roughly an hour and a half after launch. The 3rd stage could be fairly small with a modest propulsion system, it only has to boost payloads from LEO to GTO etc. so it could be fairly low cost and easy to develop. Plus, it wouldn't be used at all on LEO missions.

But that still leaves all the rest. Most likely they add draco/superdraco thrusters to the 2nd stage along with landing legs and thermal protection systems. The thrusters alone might be enough to provide attitude control authority through re-entry and landing. Adding all that mass and complexity will increase the cost of the 2nd stage, while, for the most part, lowering the payload.

But on the plus side, even with significant increases in stage cost and mass they should be able to bring the per flight amortized hardware costs due to the 2nd stage down from over $10 million to under $2 million.

With first stage reuse alone they should be able to lower their costs by up to a factor of 3 or so, with second stage reuse they should be able to lower that by a further factor of 2 or more, making it possible to provide launches for under $10 million.