Learn Turkish: O köpek çok kızgın, ona dokunamam. - That dog is very angry, I cannot touch it.

The difference is just one letter, but it changes the meaning completely! dokunamam means "I cannot touch" (inability). If you remove the -a- in the middle, you get dokunmam, which simply means "I do not touch" (regular negative aorist tense). That single -a- before the negative -ma is what adds the "cannot" meaning.

How exactly is dokunamam broken down into suffixes?

It starts with the verb root dokun- (touch). Then we add the inability suffix -a (cannot). Next is the negative aorist suffix -ma (which drops its usual 'r' for first-person "I" and "we"), and finally the first-person ending -m. Put together: dokun-a-ma-m.

Why does the sentence use ona (to it) instead of onu (it) when saying "touch it"?

In English, we just "touch it", but the Turkish verb dokunmak requires the dative case (the "to / towards" direction). You must literally say "touch to it", which is why we use ona instead of the direct object onu.

Is the word kızgın (angry) related to the verb kızmak (to get angry)?

Yes! kızmak is the action of getting angry, while kızgın is the adjective describing the state of being angry. In Turkish, the suffix -gın (or -gin, -kın, etc., depending on vowel harmony) is often added to verbs to turn them into adjectives.