Learn Turkish: Israr etme, gitmiyorum. - Do not insist, I am not going.

In Turkish, many verbs are created by combining a noun with a helper verb like etmek (to do or to make). Israr means "insistence", so ısrar etmek literally means "to make insistence" or "to insist". Here, etme is the negative command form.

Does etme mean "don't do", or is it the dictionary form of the verb?

Here it means "don't do". The suffix -me is used to make negative commands. It looks exactly like the suffix used to turn verbs into nouns (like "doing"), but you can tell from the context of this sentence that you are giving a direct order: "Don't insist!"

The verb gitmek usually changes to gid-, like in gidiyorum. Why does it stay as a "t" in gitmiyorum?

The "t" in gitmek only changes to "d" when the very next letter is a vowel. In this sentence, the verb root git- is followed by the negative suffix -mi-, which starts with the consonant "m". Because it is followed by a consonant, the "t" doesn't change.

The negative suffix is usually -me or -ma. Why is it -mi in gitmiyorum?