As a rule, gerunds should be used like their verbal counterparts. If the verb is a phrasal verb, then the gerund should be constructed accordingly.
So your example "Thinking of you makes me happy" is correct, because you'd use "of" if you rephrased this with "think" as a verb:
When I think of you, it makes me happy.
However, this is incorrect
He has finished washing of the dishes.
because you'd say "I don't want to wash the dishes," not "I don't want to wash of the dishes." That last sentence makes no sense in English.
The Inevitable and Rarely Used Exception
You knew there was going to be an exception, right?
In a few cases, using the gerund + of is correct when it is important to emphasize the specific act the gerund describes. It also sounds extremely formal to an English speaker; it is not a construction one would use in everyday speech except as a joke.
Because this construction refers to a definite act, it must take the definite article "the." The phrase "She enjoys reading of poems" would sound odd to an English speaker, but "She enjoys the reading of poems" is perfectly correct, though having an elevated tone.
And because "the reading of poems" draws attention to the act and not the person performing the act, it can imply the reading of poems by others in addition to her own reading of poems.
A few more examples:
The signing of the Declaration of Independence actually took place on July 2.
The naming of new Nobel Prize laureates is always exciting.
I love the turning of the seasons.
Notice how the use of "the" refers to a specific act or event. The sentence
Signing the Declaration of Independence actually took place on July 2.
does not make sense because without "the," it implies a habitual action, not a specific act or event.
Humorous use of this construction usually makes an insignificant event sound more important than it really is:
Now is the time for the washing of dishes.
I humbly invite you to join me in the drinking of beer.
I don't know a name for this. There might be one, but there isn't a one-word name for everything you could possibly do. I'd call it "one subject with multiple verbs".
As to whether it's legal: yes, certainly. It's very common. It's an effective way to express a connection between past and future events. "The committee started its work yesterday and will finish tomorrow."
Best Answer
Gerund v. Participle
I think you're confusing gerunds and participles. While they look the same (note: in Old English, they were different), they function entirely differently.
The first set of -ing words you give are not gerunds, but participles. These are verbal adjectives that modify a noun or pronoun.
Gerunds, however, are verbal nouns. They do not modify other nouns or pronouns.
Make and Have
For make, you use an accusative plus infinitive without "to". All of your examples are correct. It's she made me do it— no participles needed!
For have, the sentences are actually awkward English. It would be better to switch it around: