Good post. I've been thinking about this as well, and my general idea is similar to yours.
However, I'd scatch #2 for the reason Argos mentioned. Also, you can't discount the possibility of a large but undifferentiated object: Callisto, for instance, is just barely.
I'd definately set the lower limit of a planet to where it's large enough to become spheroid. Now, there are a few overlaps here: Vesta, Pallas, and Proteus are all larger than Mimas, yet they are irregularly-shaped, whereas Mimas is spheroid. Thus, Mimas would count as a planet, but the other larger ones wouldn't. But even much larger objects such as Iapetus and 2003 EL61 aren't quite spheroids either. The latter could fit as a highly oblate spheroid, but I have to admit I'm at lost about Iapetus. For now, let's discard that one and worry later.
So, if it's a spheroid, it's a planet. If it's not, it's an asteroid, comet, KBO, debris, etc. The upper boundary is whether it burns deuterium or not. I don't care about orbit, composition, mass, any of that.
What IS an important question, IMO, is whether to allow moons of planets as planets themselves. Are we going to call Ceres a planet but not Ganymede just because of what kind of body they orbit? This is a tough question for me. It might be confusing to hear in the news that "Neptune has another planet." Or would it? I think calling spheroid moons as planets would help seperate them from the other heaps of rubble also called moons (which I think should be called moonlets or moonoids instead). Perhaps it could go like this:
All spherical moons are planets, but not all planets are spherical moons.
Just like how all squares are rectanges, but not all rectangles are squares. Might sound confusing at first glance, but in a brief summary of, say, Titan, it would be introduced: "Titan, a satellite of Saturn, is a very interesting planet to say the least." I don't see why this couldn't work. Afterall, we already refer to the moon as one of the terrestrial planets. So, I accept spheroids orbiting planets as planets too...moon or satellite would just describe an important orbital aspect of the planet.
A non-fusing spheroid not orbiting a star can be called a rogue planet. I don't see what the debate is about that type of object not being a planet; this is going back to the arguement that where the object is or what it orbits should not matter.
If we call moons planets, this obviously makes the debate of whether a pair is a double planet or a planet-moon system a mooter point. Since my definitions allow the term moon to still be used to specify where the planet orbits, it still needs to be defined. Maybe depending on where the central point of gravity is (if it's on the surface of a body, that body is the parent one...if it's in between both objects, they're double planets). I haven't read much about this area so this may be stupid. But I'm sure something can be reached.
Finally, adjectives to group planets based on location and structure is allowed, but not a necessity. Mercury, Venus, Earth, Luna, Mars are terrestrial planets...Jupiter, Saturn, Uranus, Neptune are gas planets...Triton, Pluto, Quaoar, Orcus, etc are Kuiper Belt planets. Sedna could fit either there or as an Oort cloud planet. As for the planets orbiting the gas planets, most of them seem intermediate between terrestrial and Kuiper, so how about Intermediate planets? Ceres, a spheroid having an ice mantle like the Galileans, would also be an Intermediate planet.
Let me repeat that tagging adjectives is NOT a necessity under my classification. Sort of like how bears can be grouped into black bears, grizzly bears, polar bears...but you don't HAVE to specify what kind of bear when you scream there's one in your garage.
It's more common to just say BEAR rather than what type. Some planets may be debateable or hard to put in a group...in this case, debate is more acceptable since it won't make the difference between planet or no planet.
To sum up:
-A planet is simply a gravationally relaxed spheroid and a non-fusor.
-A planet in orbit around another planet may also be called a moon or satellite.
-Planets can be subdivided by location and/or structure (terrestrial, intermediate, kuiper belt, gas).
-A non-spheroid object is called a planetoid.
-A non-spheroid object in orbit around another planet may also be called a moonlet or moonoid.
-Planetoids can be subdivided by location and/or structure (asteroid, centaurs, kuiperoids, comets).
So the planets in our solar system would include the 9 traditional ones, plus all the spheroid moons, the spheroid KBOs, and Ceres. *Edit*Since I'm trying to stay technical with no ifs/ands/buts...Iapetus and maybe EL61 are planetoids under this scheme because they're irregular. I'm not going to try to make up an excuse as to why they should be exceptions just because they're too big. But they may end up being the tomatoes of the planetary business and be regarded as planets anyways.
For those who are going, holy crap no way...I'll close my post with something that an older (but still smart) person told me. She said she remembers when there were two dozen or so elements, and how they memorized them in school. I guess if something similar to this or what the topic poster suggested takes hold, there won't be any fun memorizing anymore, but that's just like what happened with the elements. This happens in science all the time, and it's time to expand planet classification to account for our new understanding. There's no way I can look at our solar system and say there's 9 planets anymore...the line has to be drawn somewhere less arbitrary. Spheroid is the least arbitrary so it's the best IMO. And it's probably time we start accepting moons as planets too. I never learned about Europa, Io, Titan as a kid (I was a kid in the 90s, so yes they were important), just because they happened to orbit a planet and not the sun. While my suggestion results in a ton of planets...the definition itself is extremely simple. It'd also be easy to maintain as we discover new objects around the fringes of our solar system, and allow the news to say "New planet" rather than the stupid "New planet-like object".
But hey, it's certainly open for debate.