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.
The gerund refers to the act or process of doing something - the activity itself and nothing further.
The -ion form of a root can mean the act of doing something, but usually leans toward meaning its result, effect, or manifestation - something that persists or evidences after the activity.
Why don't some verbs have "-tion" nouns?
This may be a better question for https://english.stackexchange.com, but briefly looking into it, -ion comes from Latin and is generally used with words of Latin origin. Non-Latin words - such as those that are part of the Germanic "core" of English (e.g. all the irregular verbs) won't work with -ion.
Best Answer
Consider the following:
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.
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 did not see them off:
It was impossible to see them off: