We don't know whether shuttles at this timepoint have shields, really. TOS shuttles didn't necessarily appear to have any, unless such were specially rigged ("Immunity Syndrome"). STXI shuttles didn't necessarily appear to have any, save for Scotty's very special hot rod. And the ability to take coronal heat was a groundbreaking late 24th century invention in "Suspicions".
(Did ST:ID feature explicit references to shuttle shields, though?)
Think of it as the way the flying fish move. Jump out of the slow medium, glide through the fast one, and return to the original medium is assured at the end. You use a machine to plunge your pattern into phased state, where it can be shot towards a target, and eventually it resurfaces into its more stable, non-phased state, hopefully at the target.
A machine the size of a briefcase doesn't sound implausible as such if it's used at the starting point to plunge Harrison into phased space. If it's used at the destination, then it truly has interstellar range and becomes implausible - but if it places people in phased state and this state then takes care of the rest, it's not really a range issue any more and you don't need to insist on machines the size of large asteroids.
The writers of this franchise are prone to do things that they think will look cool without really thinking them through, such as wasting an escape pod instead of beaming Kirk to the brig of the outpost so he can meet a monster.
Actually, I think they covered that one, more or less - the transporter had just killed Amanda, and might not be high up on the repair list of Olsen's successor.
A Volcanologist did an analysis of the science on Trekmovie.com. Basically entering a planet's atmosphere is typically 500+ degrees hotter than a volcano so the problems suffered by the shuttle were indeed silly. However, they point out that volcanic ash can indeed be problematic so replace heat with ash and the scene works fine.
Well, except that it isn't clear why they can't just scan the volcano's interior and beam a device in with a timer, or why the volcano threatens the planet rather than just the local population, or what the device is actually supposed to do, since capping the volcano will just lead to pressure increasing until the volcano explodes with even greater force. Wibbly wobbly timey wimey.
The way I interpret transporting is that you are quantum entangling the transportee's molecules and then phasing the individual into a subspace pocket, replacing them with the matter stream, which is the corresponding entangled energy from the subspace pocket (this overcomes the kill and clone argument which Trek has trouble with and can explain Realm of Fear or Thomas Riker).
It is the annular confinement beam that keeps all the qunatum linked energy in the right place. This is necessary because once the beam is switched off, the person automatically phases back and if that happens you really want all your bits in the right place (see TMP). So a transporter is sending quantum linked subspace energy on a subspace carrier wave to the desired co-ordinates.
The problem I have with transwarp beaming is that something has to maintain the confinement beam until you reach the destination. This is why I think the concept works if you use relay stations (like the stargate bridge) within your own space, which would include a receiving pad (just in case one of the relays malfunctioned) but the pad would have to be expecting you to pick up your signal and extend its own confinement beam to meet yours. I can't see how it works if you are beaming into enemy territory without some further explanation.
I understand that TNG featured the technology but I never really saw a satisfactory answer to how it worked. Gary Seven's transport looked a bit different. If you can create a wormhole, you can just step through it but the energy required would be pretty amazing if you are scrunching space instantaneously across light years. That would seem to me to be the bar to this working for Federation level races.
Having said that, scientists acknowledge that warp drive would require more energy than anti-matter could supply. I am willing to hand wave that away if they keep the speeds sensible. If they keep compounding the issue it starts to annoy.
As far as Amanda's death goes, it isn't really a plus in favour of transwarp beaming. It might imply that beaming up requires a localised quantum scanner to help entangle all your atoms and that is much harder if you are not transporting pad to pad. I like that limitation.
It's in a different ball park to beaming across light years though. Planets are moving, space is moving, the sending pad is moving, and from what we know of our own planet, that movement fluctuates depending on our sun and other objects in the vicinity. I still don't see how you can reliably transport to co-ordinates 90 light years away. If you can't scan, not only the landing site, but the movement of the planet in relation to the place from which you are sending before you initiate transport it's pretty much a death sentence.