Your language seems pretty Democrat-biased to me, but I suppose that's par for the course around here. (Which isn't necessarily a judgement. I don't like the Republican party either).
The way I see it, you can sum-up the difference succinctly: Republicans place the personal over the social, while Democrats do the opposite. Most things follow from there. (Corporations, in this view, are indeed an extension of the individual). (The Republican emphasis on religion and "family values" is an enigma to me, I don't know where it comes from, and it seems perverse in terms of my model. It breaks a lot of things. Libertarians are what I think Republicans should be).
Thus, (in general) Republicans emphasise the individual's responsibility in getting an education and making money, while Democrats emphasise society's/the government's responsibility to support individuals in this.
Democrats insist that society must support itself through taxes on individuals, whereas Republicans insist that society leaves individuals (and corporations) alone, allowing them to support themselves.
Republicans demand that health and safety matters are left up to (presumably intelligent, responsible) individuals to take care of, while Democrats demand that society (/governement) protects individuals both from risks, whether posed by others (eg gun-control) and by themselves (eg environmental or food regulation).
Republicans tend to be less concerned by social inequality, including racism, sexism, etc, whereas Democrats tend to be quite concerned by these, but less concerned about telling people how to think/speak/behave/raise kids (in terms of protecting the rights and freedoms of society (racism or sexism is more a social issue than an individual one).
And so on...
As I mentioned above, the way I see it, Republicans should be in favour of Gay Marriage and stuff like that. I can't say why they aren't, and perhaps my simplification is a little too simple. I see it more as an anomaly than as a theory-breaking flaw.
[If I were American, I would very happily vote for Gary Johnson.]