I can only speak for myself, and there are many ways to answer this question; but my short answer is this:
The only one that does not sound idiomatic (to me) is 3a.
?3a - I don’t like basketball, I’d prefer playing football.
All the others sound idiomatic. Later, we'll see how to make 3a sound idiomatic.
I really dislike one or two-sentence contexts, because they rarely model anything from real-life discourse. They are abstractions which allegedly serve to exemplify a given condition, but this rarely works.
For instance, I find the most natural way to understand these statements is as an answer to some question (Q1) such as:
Q1 - We're all going to play basketball, do you want to play (with us)?
Note that the use of today in such a context is not necessary, since it is understood.
And all the expressions with would in them are quite satisfactory, except 3a. So all the following work as an idiomatic answer to the question, with or without today. That is, you could add today to 4a and 5a, and remove today from 3a, 4a, 5a without a problem.
4a - I don’t like basketball, I’d prefer to play football.
5a - I don’t like basketball, I’d rather play football.
3b - I like football, but I’d prefer playing basketball today.
4b - I like football, but I’d prefer to play basketball today.
5b - I like football, but I’d rather play basketball today.
Those are all natural souding and quite satisfactory just the way they are.
I don't know what it is about the would, but it makes these statements perfectly natural as an answer to the proposed question. If I had to speculate, it might be the case that the structure is similar to a "fulfilled conditional," namely,
6 [Since you've given me the choice,] I'd prefer or I'd rather...
However, the statements without would also work as answers to the same question. But instead of stating it the manner of a "fulfilled conditional," they state it in the manner of what is always true:
1a - I don’t like basketball. I prefer playing football.
2a - I don’t like basketball. I prefer to play football.
The phrase You know that could easily go in front of both those sentences. Thus, showing that the first sentence is true all the time (given the contraints of one or two two sentence contexts).
1c - (You know that) I don’t like basketball. I prefer playing football.
2c - (You know that) I don’t like basketball. I prefer to play football.
The prefer phrase states what the person likes all the time., thus:
1d - (You know that) I don’t like basketball. What I like is playing football.
2d - (You know that) I don’t like basketball. What I like is to play football.
As for the B-Statements that did not sound idomatic to you, they are just fine
1b - I like football, but I prefer playing basketball today.
2b - I like football, but I prefer to play basketball today.
They are quite satisfactory, absolutely no problem. They sound perfectly natural, with or without You know that in front of them. Since you included today with them, they especially fit as an answer to the proposed question.
In other words, I do not agree that whereas 'prefer' is used for general topics
Now for the odd phrase out.
First let's restate the question:
Q1 - We're all going to play basketball, do you want to play (with us)?
and the problem child:
?3a - I don’t like basketball, I’d prefer playing football.
Let's try you know that:
?3c - (You know that) I don’t like basketball, I’d prefer playing football.
Hmm. Not much better (to me).
What about the d-form without you know that:
3d - I don’t like basketball, What I like is playing football.
That works.
I think 3a is probably normal and idiomatic for some people. But to me, it cries out to be expressed as in 3d, above, or as in
3e - I don’t like basketball, I’d prefer playing football today. (specific)
3f - I don't like basketball, I'd prefer playing football anyday. (general)
Or use an explicit comparison:
3g - I don't like basketball. I'd prefer playing football to basketball (today/anyday).
