Learn Hardcore Japanese: oyasirazu wo nuita ato, haguki ga hareta node, yukkuri kamimasu. - After my wisdom tooth was extracted, my gums swelled, so I chew slowly.

Breakdown of oyashirazu o nuita ato, haguki ga hareta node, yukkuri kamimasu.

Why is を used after 親知らず, but が is used after 歯茎?

Because the two nouns play different roles in the sentence.

親知らずを抜いた
Here, 親知らず is the direct object of 抜いた (pulled out), so it takes を.
歯茎が腫れた
Here, 歯茎 is the thing that became swollen, so it takes が as the subject of 腫れた.

So the structure is basically:

This is very common in Japanese: one clause can use を, and the next clause can use が, depending on the verb.

Why is it 抜いたあと and not 抜くあと?

Because あと means after, and when you say after doing X, Japanese usually uses the plain past form before あと:

So 親知らずを抜いたあと means after I had my wisdom tooth removed / after pulling out my wisdom tooth.

Using 抜くあと would sound unnatural here.