Obviously the main purpose of this episode was to introduce the Doctor's new travelling companion (Martha Jones), and I think it did a good job with that. She came across as intelligent without being infallible (or having encyclopaedic knowledge) and aware of current events, so that's a nice contrast with whatshername from the Christmas special ("good luck, Lance!"). I think it can be tricky to get the right balance, because if you have other characters standing around saying "Gosh, that's an amazingly insightful comment" then it may seem self-congratulatory on the part of the writer, but this episode managed to avoid that trap. It's also good to see that Martha has a sense of adventure, by enjoying the experience of being on the Moon and being willing to go outside the building.
There's an issue of Lucifer where someone asks Jill Presto whether she's a doctor, and she says "Not in these heels, honey" (paraphrasing from memory). It may seem slightly odd for Martha to be wearing high heels, but some of the female doctors I work with do the same thing, and at least they're not stillettos.
Since they discussed the question of air, it's slightly odd that they didn't bring up gravity; I'll be charitable and assume that it was taken care of by the force field dome (or bubble). As for electricity, I'm guessing that the hospital has its own generator (based on where I work). Water would be a bit more tricky, since the pipes were all cut.
Since people on Earth-DW seem to have taken alien races pretty much in stride, I think that the best approach on the Moon would have been to close all the curtains. Out of sight, out of mind... After that, the doctors would be able to just stick everyone back in bed and resume the normal routine to settle things down. (Admittedly, this was my plan before the rhino aliens turned up, which disrupted the routine a bit.)
One of the weak points in Star Trek has always been the universal translator, because simultaneous translation simply doesn't work across different languages. (Or at least, there will be some cases where it's feasible and many where it's not.) For instance, "le chat noir" in French corresponds to "the black cat" in English, but if you translated it word by word you'd get "the cat black". (There's a similar problem with their communicators, which would only work if they broadcast to the entire ship, but I digress.) By contrast, I think that this episode handled it nicely, by just waiting until the machine had heard enough of the doctor's speech to identify his language.
I'll be interested to see whether Martha keeps referring to the Doctor as "Doctor", given that she's one too. For that matter, even "Mr Smith" is quite formal (if he's using her first name), so maybe she'll start calling him "John"? As a related issue, it's slightly odd that he referred to himself as a Time Lord when giving his species, since I'd think of that as more of a job description; has anyone referred to Gallifrey by name since the relaunch with the 9th Doctor?
The only real flaw I saw was when Martha tried to do CPR. More specifically, what the hell kind of ratio was that?! She was doing 5 chest compressions to every 1 breath, and I've never come across that for a single first aider treating an adult casualty. (Bear in mind that this was before she remembered the "two hearts" thing, so she wasn't allowing for that.) For the record, the current (2005) guidelines from the Resuscitation Council say that for an adult [human] casualty you should alternate 30 chest compressions with 2 rescue breaths. Previously, that ratio was 15:2. When I did lifeguard training at school (1992), we were taught to do a ratio of 5:1 if you had two first aiders working in tandem, but that method hasn't been taught in several years. (The rules are slightly different for babies and children, but I'd recommend that everyone should take a one day first aid course to get the hang of the basics.) So, I'll assume that she was skiving off when they taught that at medical school... Still, one glaring mistake per episode isn't bad (as compared to Torchwood for instance), so I'll let it slide; I just think that it's an important point to note in case anyone is in a corresponding situation in real life.
Thinking about the Doctor's travelling companions, most of them have seemed to have signed up by accident (e.g. Tegan trying to make a phone call at an airport and being stuck in the TARDIS for a few years). As the Doctor has learned to control it a bit better, he's been able to offer people an informed choice, but there is still a problem of dropping everything to head off over the horizon. One issue would be family and friends, so this episode helped by showing that she could probably do with a break from her relatives. (I'm not quite sure whether that effect was intentional, but it works for me.) The job commitments issue is a bit more tricky, so if I was offered the chance to go then I'd have to say "no". Even if I could come back exactly when I left, there would still be logistical problems, since my memory would be a bit rusty about various documents/projects that I'm currently working on. Having said that, I'd certainly be happy to make a one-off jaunt across the universe.
If you were offered the chance to drop everything and go off in the TARDIS, right this minute, would you go?