I'd run two conduits instead of one
You'll want to run two conduits out to this shed instead of just one. Why? At some point, you'll probably want a hardwired data service to your shed, and running a separate conduit for data makes that much easier. As to conduit sizing? A pair of 1.5" PVC conduits provide ample room for most things you'd want to do, including Harper's 2-2-2-4 mobile home feeder. The data conduit can terminate in bell-ends inside the walls of the house and shed, while the power conduit terminates in the shed panel as Harper describes, and transitions to ENT ("smurf tube") going back to the house's panel when it enters the house walls.
GO BIG OR GO HOME
As far as that panel in the shed goes, I echo Harper's suggestion of getting a panel that's convertible from main lug to main breaker. This pretty much automatically puts you in a 12-space or greater panel if you don't want to burn spaces on a backfed main breaker later, but I'd go larger yet as panel spaces are cheap. In fact, a 24-space or 30-space, 125A, convertible main panel would be not at all out of place here, and using a NEMA 3R (outdoor/weatherproof) panel means that you don't have to allocate clear working space for it indoors.
Just remember to get and fit the correct grounding bar for your panel if it doesn't come with one, and also remove the green bonding screw or strap so that neutral and ground are properly separated! You'll also want to install a pair of 8' deep ground rods driven 8' apart and connected to the panel grounding bus with 6AWG bare copper, unless you are putting an Ufer (concrete-encased electrode) in at your shed. That takes care of the grounding electrode system for your shed, which'll be required as soon as you install more than one throw of disconnecting in your subpanel anyway.
As to what goes in those conduits...
While I don't think the voltage drop situation is quite as bad as Harper makes it out to be, you still will want to take it into some account by running 240V to the shed. This means that you'll be using a two-pole breaker in the subpanel as your shutoff, and four stranded THHN wires instead of a UF cable inside the power conduit. However, over this distance, for a 240V run, 12AWG is adequate, with 20A breakers at both the main panel and the shed subpanel; if you want to, you could bump up to 10AWG instead, but you shouldn't need to go further than that.
As far as the data conduit goes, you'll want to pull fiber through it instead of outdoor-rated 4P UTP cables. This has nothing to do with bandwidth, though, and everything to do with lightning. You see, when copper communications cables are run out doors, the NEC requires what's called primary protection at each end of the cable, which is a significant cost and time adder to the project since you can't buy such things at Home Depot. Nonconductive fiber optic cables don't need any of that, though, just a pair of switches with fiber transceivers in them. Of course, if you have adequate WLAN signals at the shed, you don't need to pull the fiber now; in that case, the conduit simply becomes a future provision, which is fine.
And keeping things nice and cozy in there
While your solution of a small space heater (such as an inexpensive 120V baseboard) and a window A/C is not inadequate for a shed of your size, the more efficient solution would be to use a window or through-wall mounted heat pump. This kills two birds with one stone, and provides you with a more efficient heat source than straight electric heat, while still having electric backup heat available to fill in shortfalls in heating.
Notably, while many TTW and window-mount heat pumps run on 230V, the through-the-wall mounted Amana PBH092 runs on 115V and is available for a reasonable price through a variety of online suppliers (Home Depot actually overcharges significantly for this model). You'll need the matching PBWS01A wall sleeve, of course, and that does drive the price up slightly, but it's also not a hard part to find, and lets you upgrade to a higher-capacity unit later when you fit the full subpanel if you wish more grunt from your shed-office's HVAC system.
Finally, instead of throwing good energy dollars at a bad envelope, I'd construct the shed with good insulation and air-sealing now, as it's far cheaper to get it right the first time. A good air/water barrier (fluid applied or fully adhered) over the sheathing, along with good flashing details, is essential to keep hot Texas air, or rain for that matter, from leaking in, and I'd apply 2-4" of rigid insulation board over the top of that to provide a good, continuous thermal barrier as well.