All three sentences are correct, but i dont understand why we should not use infinitive in first sentence after rather than? What basically rule is ?
I decided to skip lunch rather than eat in the cafeteria again >>> this sentence can be compared to >>>
a) I like to eat and (i like to)
Though "to" is not mentioned explicitly, its inherent in the sentence.
I think grammatically it will not be wrong to use "to", but it's not the most concise sentence..!!