You can always use "I'd" rather than "I would" in speech or in most writing any time you want no matter what comes after it. In very formal writing, it's best to avoid most contractions. Some verbs can be followed by the gerund (-ing form) or the infinitive (to + base verb) with no change in meaning, and "like" is one of them. We don't usually use "fancy" as a verb in American English, but I think it's normally followed by the gerund when there's another verb after it. Perhaps one of our British colleagues can confirm if that's the case.

Hi Jonathan British colleague here! Yes, that's correct "fancy" in this usage would be followed by the -ING form of the verb. For example: "Do you fancy comING out tonight?" All the best Jonny