Yes, the NEC lets you do that
The NEC has absolutely no problem with letting you have your very own overhead cable run. This is called messenger supported wiring, and can be done using by lashing a UF cable to a separate messenger wire, as per NEC Table 396.10(A). However, a more conventional approach to this is to use an overhead multiplex cable, much like the ones used for service drops, although with an extra wire included since you're running a feeder, not a service. It's also somewhat cheaper to use the multiplex cable, as a 6-6-6-6 "Chola" quadruplex costs about the same per foot as 12/3 UF-B, and the latter requires a separate messenger wire.
So, we'll start with that for our 65' run. Since we're dealing with an "open multiconductor cable", the NEC 225.18 minimum clearance of 10' from the finished grade or sidewalk to the bottom of the cable sag applies, provided you aren't crossing a driveway. (If you do have to go over the driveway, then you have to increase the clearance to 12'.)
So, given our quadruplex cable from above at 0.162lbs/ft and about 0.625" diameter, the sag and tension calculations from the NESC as restated on pp. 33-35 of Southwire's installation manual, and the fact that we're dealing with "Heavy" ice loading as per the NESC's ice/wind loading figures and your location, we get an ice loading of 0.698lb/ft, a wind loading of 0.542lb/ft, and a resulting cable load weight of 1.32lb/ft. From there, we can use that, your 65' span, and a 1.5' sag to get a loaded tension of 465 lbs, which is well under 50% of the 1190lb load rating on our "Chola" cable's integral 6AWG ACSR messenger. We then add 6" to this to provide a margin for heat sag.
As to getting up to 12' or more above ground...
Obviously, you'll need a mast with a weatherhead at the shed; this can be done using a single stick of 1.25" RMC with some 1.25" two-hole straps, a 1" nipple, a 1.25" to 1" reducer bushing, and a 1" LL or LR body at the shed end of the circuit, along with a 1.25" threaded or clamp-type weatherhead, 4AWG Al XHHW-2 for the in-mast wires, some Burndy AGSKIT2 or equivalent dual-rated Al9Cu aboveground splices for the hots and neutral, a clamp-type wireholder for anchoring the quadruplex cable's integral messenger to the mast mechanically, and an Ilsco GTA-2-2 and Ilsco AGC-1 or equivalent dual-rated tap connector and pipe clamp to allow a 4AWG Al jumper to be used to bond the messenger to the mast. Don't forget to form a drip loop in the conductors extending out of the weatherhead before splicing them to the overhead wires!
At the house, you can either use a similar mast and weatherhead, or SER cable formed into a gooseneck along with a beefy eyebolt (remember that loaded tension from before, as well as the angle the drop is making) to secure the messenger to a stud. Either way, you'll need the same parts for splicing the mast or SER wires to the overhead wires as you used at the shed, and you'll also want to remember the drip loop here, too.
As to ampacity, we're in luck: the 4AWG Al mast wires are the limiting factor here, permitting 65A at 75°C. This is because we can run the overhead quadruplex at its 69A 90°C rating thanks to the 110.14(C)(2) separately installed pressure connector provisions applying to this sort of run. A 60A or 70A breaker can be used for this feeder if you're putting a panel and grounding electrodes in at the shed; if you wish to dispense with said panel, you're limited to a 20A, 2-pole breaker so that you can treat this as a (fat) multi-wire branch circuit instead.
And wiring up the shed...
You have two options as to how to wire the shed; either you can put a simple non-fused pullout (air conditioner) disconnect in and treat the circuit to the shed as a branch circuit, or you can put a panel and grounding electrodes in at the shed and thus have a full feeder running to the shed. The disconnect is somewhat simpler and cheaper than the panel, but severely constrains how much power you can provide at the shed compared to the panel approach, as mentioned above. You'll also need to provide a 3-port, 14-4, dual-rated insulated mechanical connector for the neutral connections at the disconnect if you go that route since an A/C disconnect only provides a single, bonded block. From there, you'll have two 20A, 120V branch circuits; one for the lights and receptacles, the other for a 120V heat pump.
If you do decide to go with a panel...
If you go with a panel, you'll want to go big here; in fact, a 24- or 30-space, 100 or 125A, NEMA 3R (outdoor rated) main breaker panel is not at all out of place in this application, as the main breaker simply provides the outbuilding's disconnecting means. You'll need to fit a separate grounding bar of the appropriate type for your new panel and land the incoming grounding wires on that atop pulling the bonding screw from the neutral, of course, since this is a 4-wire feeder.
Speaking of grounding, that panel means that you'll need to take some 6AWG copper and run it from the grounding bar, out the bottom of the panel, to a pair of 8' ground rods driven 8' or so apart. This provides a grounding electrode system for your shed so that wayward natural electricity (such as lightning-induced transients) can get returned to Mother Nature, while the feeder equipment grounding conductor returns errant utility electricity back to the utility via the neutral-ground bond in your main panel.
Note also that if you go with the outdoor panel route described above, you can simply use a 1.25" bolt-on hub atop the breaker panel instead of using the LL or LR body, reducer, nipple, and locknuts to bring the mast into panel from the side.