A really sneaky fallacy, or just dishonest arguing?

“Women (C) are more emotional people (A), men (B) are just not sensitive enough”

Seems like an opinion, though I'd like to ask what the person means by 'more emotional' - do women simply express more emotion, or experience more?

There's also a misunderstanding on your part - someone saying men aren't sufficiently sensitive isn't suggesting they have no feelings at all. 

“Speaking proper English (A) is very British (C), no one abroad (B) can really speak proper English”

"No one abroad can speak proper English" - I'm guessing they mean, no foreigners can speak English very well. This, taken literally, is amazing familiarity (though it is probably a hasty generalization extrapolated from prejudice, or the experience of a non-native speaker struggling with their English).

“All racists (A) vote Republican (C), Democrats (B) can’t be racist”

There is more amazing familiarity at work as the person asserts all racists vote Republican. "Democrats can't be racist" is just a false claim (where's the mechanism that stops them from being so?)

“Middle Eastern countries (C) are the ones that facilitate terrorism (A), westerners (B) also sometimes carry out violence but that’s not the same”

You can question the claim that Middle Eastern countries 'facilitate terrorism'.

More pressingly, though, I'm struggling to see the underlying logic - first we're talking about Middle Eastern countries, then we talk about Western individuals. So is the claim about countries promoting terror, or individuals from those countries engaging in it?

I would be grateful to anyone who can review my thinking on this.

“Women (C) are more emotional people (A), men (B) are just not sensitive enough” 

Using your initial formulation, it would look like this,

Some more emotional people [A] are men [B]
Some more emotional people [A] are women [C]
I consider all more emotional people [A] are women [C]

There are problems with this in that it uses two type "I" ( some A) logical forms to infer a type "A" logical form ( all A). I don't think it is logically valid to move from some to all in that way (I'm happy to be corrected on that if someone knows different).

The sentence above, however, seems to formulate as,

Some women [C] are more emotional people [A]
Therefore  men [B] are not sensitive enough

You move from one premise to a conclusion - you appear to be missing a minor premise.

Could it be,
(I've played around with the wording to hopefully make my thinking clearer)

Some women [C] are more emotional than some men [A]
Some men [B] are not more emotional than some men [not-A]
Therefore all men [D?] are not emotional enough

Some C are A
Some B are not A
Therefore D are not emotional enough

This is also logically invalid.

Even if we accept the implicit assumption that more emotional than some men [A] is better than not more emotional than some men [not-A], it introduces a fourth term ( all men - as opposed to some men ). This means it is not a categorical  syllogism and therefore is a non sequiter.