You don't beam down shuttlecraft. Their very existence in terms of the storytelling aspects of Star Trek precludes the capacity to beam something down to a planet's surface.
In the Original Series, the transporter was created to skip the use of shuttlecraft in general. Rodenberry thought having characters go to a room, step on a pad and appear on a surface, would move the story keeping people interested. Later, shuttles became a plot element but were generally skipped due to their disruption of the narrative flow.
If we stay in-universe, my suspicions say the size of the average shuttlecraft might exceed the cargo transporting capacity of even a ship such as the Enterprise. And certainly the heavy cargo shuttlecraft like the Argo would most likely be outside of the capacity of a standard cargo transport array.
As for the Argo buggy, I don't see any particular reason to not beam it down unless it was larger than the transport area (or mass allotment) allowed by the transporter. I think the entire scene was created for a purpose (in my opinion) to have a great outdoor, on-planet adventure thing the viewers could relate to. Something non-holographic that reminded the viewers the crew could exist and do physical things in the real world.
Cargo transporters were larger-scale versions of personnel transporters and were optimized for the transport of inanimate objects. These transporters were adapted to handle massive quantities of material.
In case of an emergency, cargo transporters could be reset to quantum-level mode, making lifeform transport possible. One reason for such a reconfiguration was to expedite an evacuation of personnel. (TNG: "11001001")
Cargo transporters were mostly found inside the cargo bay of a starship or space station. On Level 97-C of the Spacedock-type Starbase 74, there were four cargo transporters. (TNG: "11001001")
Dedicated cargo transporter platforms used by Starfleet in the 24th century typically featured one large circular or oblong pad.
Both of these images show cargo transporters onboard the Enterprise. Neither are large enough to consider moving something as large as the Argo, a warp-capable heavy transport shuttlecraft.
We have never seen an entire starship being transported, except by the technologically advanced Voth, who transported the entire starship Voyager into its cargo hold.
The Voth city ship was massive and extremely powerful in comparison with Starfleet vessels – the Voth were able to beam the USS Voyager into a chamber inside their vessel which could clearly house many more starships. This transport was accomplished despite Voyager's shields being raised. The Voth were also able to take command of the ship's computer, shut the majority of systems down, and take primary power offline. The dampening field used to create this effect managed to render tricorders, communicators, and phasers useless.
UPDATE: In addition to the slight change in the question, I will address another issue, size of the transport area.
As to the transport-ability of shuttlecraft: Some shuttles can be transported, some cannot. Shuttles vary widely in size, so it is technically possible at least some of the smallest could indeed be transported by a cargo-size transport pad.
There has been some controversy about site-to-site transfer and the ability to transport oversized objects. A site to site does not change the size of the transport area. The object still has to fit within the transporter window to be calculated into the transport buffer. This is about the limits of the transporter and the technology used to calculate an object in transport.
In addition, site-to-site transport cost twice the pattern buffer space, twice the transport time and energy needed. It also is used primarily for emergencies, since it probably works best with a precise sensor lock on a non-moving subject.
A site-to-site transport was a special type of transport in which an object or person was transported from one site directly to another, neither site being a transporter platform. A site-to-site transport was accomplished by first transferring the transporter's target from the site of origin to the pattern buffer of the transporter, in the same manner as the usual "beam in" procedure; instead of being routed to a transporter platform, however, the matter stream was diverted to a second site, in a similar procedure to a conventional "beam out".
Site-to-site transportation was very resource intensive. It consumed twice the energy of a conventional transport (since it was effectively two consecutive transport maneuvers), and required twice the time in the pattern buffer. It was very useful, however, when time was critical (for example, when a casualty needed immediate attention in sickbay, site-to-site transportation was almost invariably used) or other unusual circumstances. On the other hand, it was almost never used in emergency evacuations of large groups because it would effectively halve the capacity of the transporter system. --Memory Alpha: Site-to-Site Transport