'Before' and 'of' are prepositions. Other prepositions are, for example : by, with, without, in, at, on, after, for If you have a verb following any of these words, it is ALWAYS in the 'ing' form ( gerund). A gerund is a verb functioning as a noun. For example: by speaking, with reading, without looking etc. The only exception to this rule is the preposition 'to', which is sometimes followed by the gerund, but it also is often part of the infinitive form (to speak). ' 1. This should be: 'I think that after writing this entry I'm going to sleep, because it is very late in Colombia and I'm getting sleepy.' i.e. the phrase should be 'after writing', not 'after of'. 2. The gerund use is fine here, but the sentence would be better as either 'I had the idea of studying abroad a long time ago' (past simple) or 'I've had the idea of studying abroad for a long time' (present perfect). 3 & 4. These are fine. They are examples of the gerund as the subject of a sentence : 'Writing English is very useful'.

If the verb appears directly after 'of' or 'because', with no other words in between, then I believe you are correct.

"Of to sleep" and "because to write" are wrong because that would imply that "to sleep" and "to write" are nouns! Whenever you see a preposition, whatever comes after it should be a noun. Adding 'ing' to a verb turns it (or the phrase it's in) into a gerund, a verb or verb phrase that acts like a noun. In your examples, "studying abroad", "speaking English" and "writing in English" are gerunds. There is no gerund in your first example. I'm not sure why you included it.