Learn German: Ich durfte die Katze nicht streicheln, deswegen weinte ich. - I wasn't allowed to pet the cat, so I cried.

weinen (to cry) is a regular (weak) verb. To form the simple past, we take the verb stem wein- and add -te for the ich form, giving us weinte.

Why is the word order weinte ich instead of ich weinte?

The word deswegen (therefore/so) occupies the first position in the second clause. Because the conjugated verb must always take the second position in a main clause, weinte comes right after deswegen, pushing the subject ich into the third position.

Is durfte considered a regular verb since it ends in -te?

It is actually a mixed verb. durfte comes from the modal verb dürfen (to be allowed to). In the simple past, modal verbs take the regular -te ending, but they drop the umlaut from their infinitive form (the ü becomes a regular u).

Could I say Ich konnte die Katze nicht streicheln instead?

You could, but the meaning would change. Ich konnte nicht uses the past tense of können and means you lacked the physical ability or opportunity (for example, the cat ran away). Ich durfte nicht specifically means you were not given permission to do it.