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.
Consider the following:
There was no scaling that steep cliff.
Going around the mountain was the sane choice, not scaling that steep
cliff.
The first means that the cliff was impossible to scale. The second simply refers to the action of scaling as a non-choice, a thing that exists but which is rejected.
There was no reasoning with them.
It was impossible to reason with them.
no + the -ing form denies the very existence of the -ing nominal.
It is used with existential statements.
not + the -ing form accepts the existence of the -ing nominal, but negates its presence here (in this context).
He said he liked fishing, not hunting.
He thought fishing was acceptable, but said there should be no hunting.
He did not see them off:
His not seeing them off on their journey made them sad.
It was impossible to see them off:
He was in another country on a business trip. There was no seeing them off.
Best Answer
As you can see in the comments, there are many words which end in "-ing" which are not gerunds -- "filling", "thing", "icing", etc. If you're curious about the derivation of these words you can consult an etymology dictionary. Some probably originally derived from the gerund of related verbs (like "frosting"), while others simply evolved into nouns from similar-sounding older words (like "swing").
Unfortunately, while there are patterns this may not be obvious to anyone but a language expert. English a mix of several dissimilar languages and so any word can have a unique origin.