There are lots of separate questions in there: politics, physics, etc. and I won't pretend to answer all of it, but let me try to get towards what I think is the core of the matter.
How do I explain to the interested non-specialist what I do (the general field)?
My explanation actually varies a lot depending on who I'm talking to, and depends a lot on having a two-way conversation about it. However, one approach that I quite like, because it doesn't tell any lies or half-truths, it helps avoid some of the common misconceptions, and doesn't require any mind-blowing quantum stuff that I don't really have the right language to describe (apart from with maths), is the following:
I assume people are familiar with the concept of logic gates. Now, a computation is just a sequence of logic gates all wired up in a specific order for that specific computation. And you can think of a processor as something that can dynamically rewire different circuits. (Of course, the connectivity isn't dynamically changed, it's done with extra gates.) But the point is that every processor, from your basic pocket calculator up to the most powerful supercomputer, is constructed out of the same fundamental set of logic gates. (In fact, just one type of gate is sufficient. The NAND gate, for example, is 'universal' for classical computation).
Now, imagine I suddenly gave you a new type of logic gate that cannot be straightforwardly made out of your existing set of gates. That instantly gives you a low of potential to rewrite your existing software to take advantage of this new gate. That does not mean that every algorithm will be faster. Some won't benefit at all, while some specific ones might show ridiculous speed improvements. Do we know what proportion fall into each camp? Not really, until you go and work on the algorithms extensively.
What we do know in the case of quantum computation is some specific examples that are radically faster than the existing classical algorithms. Things like Shor's algorithm for factoring numbers, and Grover's search.
If it's that simple, do we have quantum computers already?
Not really. That extra logic gate is hard to build, and you can't just interface it directly with the existing logic gates; you have to build those from scratch as well in a new technology (and, really, we're still scratching around trying to figure out what a good technology to use actually is). It's a work in progress that's receiving a lot of attention at the moment. There are small working prototypes which are potentially on the verge of performing computations that we can't do classically, but they're still a long way from useful implementations of algorithms such as Grover of Shor.
Why should you care?
That depends on what your interests are. For Sophie, unless she's ultra-paranoid about her tap-and-go seamstress payment system being hacked, she probably doesn't care so much about the practical side. It's not yet clear that it's going to have a significant bearing on things. She can probably trust her payment system provider to take care of things as much as they can, and forget about it.
Why do I care? Quantum Mechanics is mind-blowing. I'm really excited not to just be a part of trying to categorise its effect, and explain certain things that are going on, but to harness that quantum weirdness and bend it to my will (cue maniacal laughter!)
Why does it get funding? Well, there are a few reasons (and I'm going to make some very sweeping statements here that are too broad). For one, the few specific killer applications that we have are really useful to specific groups. Shor's algorithm will allow government agencies to spy on communications that use existing public key cryptography systems (practically everybody). Grover's search will help with all sorts of computer sciencey problems which people like Google would really like help with).
Another reason is just to drive science on. As science progresses, whether or not the final outcome is ever realised as originally envisaged, there's all sorts of stuff comes out along the way (the classic example is the space program). Everybody wants to invest to make sure they're at the forefront of those new developments, and it's a game of chicken. Once there's major investment from one party, nobody wants to be left behind. Compared to many options, quantum computing appears reasonably accessible (while still being a huge challenge, it's not string theory) and it's amazing how much quantum mechanics has already permeated existing technology. The processor in your computer might not enable quantum computing (because it doesn't have the extra gate), but it still relies on quantum effects for its manufacture and functioning. Lasers are ubiquitous now (CD, DVD players), and their fundamental operating principle is quantum mechanical.