Here we come to the subtle difference between degree and quantity.
If you are learning to cook something from someone, and they tell you to put the pan on the stove and turn it on, you might ask them, "How hot does it need to be?"
They might respond,
"It needs to be very hot."
or they might respond,
"It needs a lot of heat."
It needs to be very hot is an expression of degree, it describes the degree to which something is hot.
It needs a lot of heat, which means the exact same thing, expresses that idea in terms of quantity. It treats hotness as a thing one can have more or less of.
Questions about degree (or questions that can be construed to be about degree) can be answered either with expressions of degree, like somewhat or with expressions of quantity, like a little bit. The reverse is not true of questions about quantity: they cannot be answered with expressions of degree.
Your first examples are all questions which can be answered in terms of degree, by (notice this!) interpreting the question not as it is literally stated (all three are boolean and the only logical literal answers are yes and no!) but as "To what extent do..."
Do you like sports cars?
Is taken to mean
To what extent do you like sports cars?
And that is how the answer can be Somewhat.
But notice that you can't actually interpret
How much sugar do you want in your tea?
as a "to what..." question and preserve what was asked. The closest you can come is,
To what extent do you want sugar in your tea?
That asks the extent of the wanting, not the quantity of the sugar, which is what was being asked. This question will not elicit answer the question "one lump or two?"
And this is why using an expression of degree, which is what somewhat is, makes no sense.
Should I give you more cake?
Can be answered with extent only as a form of wit:
Yes, you very, very much should give me more cake.
Mmmm. Cake. The question about the cake, like that about the sugar, is inquiring about quantity, and cannot be answered with degree.
Is there anything else I can do for you?
Similarly, can only become,
To what extent is there anything else I can do for you?
which while a similar question, isn't what is being asked. The real question being asked is,
What else, if anything, can I do for you?
This doesn't even take a quantity, but a list. (I.e., it doesn't return a scalar float, it returns an array of strings.) If you answer a little bit, you are characterizing your answer in anticipation of giving it, but you haven't actually answered. You will have the other person's attention to spell out what you're thinking constitutes a little bit, but you have not yet told them.