Is there a sound change from [ɡ] to [i] or [j]?

Not sure if it fits the question, but Turkish ğ sounds (to me) quite close to j. Example: büyük (big), büyüğüm (I am big).

Also Mongolian seems to just have lost a lot of g sounds. Basically whenever people would have written vowel + g + vowel some centuries ago, they just write a double vowel nowadays. And the double vowel isn't even relevant to pronunciation, only to stress.

One might suspect that the old orthography was (in the case of g between vowels) not representative of old pronunciation, but one word that has cognates in other languages (baatar, which means hero) has a consonant between the first two a's in other languages. E.g. bahadur in India, bogatyr in Russian. It is written baghatur when written in traditional Mongolian script.

[g] > [i] is a common phenomenon in Germanic/Romance and might be elsewhere. Examples are:

(Unshifted words are to the left, shifted ones are to the right):

Proto-Germanic *sagjaną > English say (cognate with German sagen)

Vulgar Latin rege > Spanish rey, Portuguese rei

Modern Swedish pronunciation of torg > /tɔrj/

I cannot think of examples in the opposite direction. The only thing that comes to mind is a different change in Egyptian Arabic, where "j" is pronounced as "g" in words such as "hijab". If you can find a previous Semitic [i] that turned into Arabic j, you would then have the desired change.

I do not believe that it is possible for [i] to become [g], plain and simple as you said. It is possible that some instances of [i] can in some context become [g], so a chain of events can turn [Cia] into [Cga] (intermediate steps being [Cja, Cɟa, Cga]), but that would not affect plain [Ci] – bia > bga but not bi > *bg. It is unclear whether you mean for these changes to be in all contexts (thus every instance of g > i), or simply in some context. For instance, in Norwegian, earlier g became [j] before front vowels (and k became ç), but g did not generally become j.

Yes, there are, and this kind of sound change is quite common.

Famous examples of this sound change are the German dialects of Berlin and Cologne, from the Berlin dialect we have the phrasal word jotwede (standard spelling jwd) that is explained as an abbreviation j.w.d. janz weit draußen (High German "ganz weit draußen" = "far away"), from Cologne the catch phrase es hätt' noch emmer joot jejange (High German "Es wäre noch immer gut gegangen", approximate meaning "What worked in the past will also work in the future").

Another example is the English language: In Anglo-Saxon, all /g/'s before front vowels were shifted to /j/ (spelled y), see also this answer for examples and counter-examples.

EDIT: Examples for the other direction are more rare and sparse, and often involve hypercorrection as a mechanism. Saxonian dialect Gung (High German: Junge "boy" comes to my mind, in this case a borrowing with hypercorrection from the Berlin-Brandenburg dialect). It is also present in proper names like Tigges from Matthias (Matthew).

There certainly is.

For example, final -y in English often comes from an older -ig (and there is often a current German cognate that ends in -ig)

It's a widespread phenomenon, one form of palatalization.

I'm not aware of any instances of the reverse change, but I hesitate to say any sound change isn't "possible"