The first 5 sentences are not actually interchangeable. A and C indicate that you do have a dog (and prefer that to having a cat), while the others simple say you would rather have a dog than a cat.
1 - I prefer being at home right now to here.
This is awkward because: 1: "I prefer being at home right now" indicates that you are at home right now, and prefer it to being elsewhere. But "to here" indicates that you are here -- and that "here" is not "at home." You're essentially trying to be in two places at once, verbally.
2 - I prefer to be at home right now rather than here.
Same problem. "I prefer to be at home right now" verbally places you at home, while "than here" verbally places you "here" which isn't "at home." Again, you are in two places at once, in this sentence.
3 - I’d prefer to be at home right now rather than here.
This sentence is fine.
4 - I’d rather be at home right now than here.
This sentence could work, but it's awkward. (Using a comma to make it "I'd rather be at home right now, than here" makes it slightly better.) It's because English doesn't like breaking up certain words, and "rather than" are words that usually want to stick together. Really, the "than here" is redundant; you could end the sentence at "right now" and be fine.
Best Answer
Yes, they're essentially the same. The first simply says that she'd spent the money on holiday if she could have, while the latter simply says that she'd choose to spend her money on holiday if it were here choice; both statements essentially say she'd choose to pay for a vacation if it were an option. They're two ways of saying the same thing.