Learn Farsi: من یک خانه کشیدم.

The dictionary form (infinitive) of this verb is کشیدن (to draw). To make the simple past tense, you drop the final -ن to get the past stem, which is کشید. Then, to match the subject من (I), you add the personal ending -م. Together, کشید + م makes کشیدم.

I've seen the verb کشیدن used to mean 'smoking' or 'pulling'. Is it the exact same verb here?

Yes, it is! کشیدن is a very flexible verb in Farsi. Depending on the context and the object that comes before it, it can mean to pull, to smoke, or to draw. Since the object here is یک خانه (a house), 'drawing' is the only meaning that makes logical sense!

Since the verb ends in -م, do I actually need to include the word من (I)?

You don't have to! In Farsi, the ending of the verb already tells us who is doing the action. Saying یک خانه کشیدم without the word من at the start is perfectly correct and sounds very natural in everyday speech.

Why is the action (drew) at the end of the sentence instead of after 'I'?

Farsi sentences normally follow a Subject-Object-Verb order. Here, من (I) is the subject, یک خانه (a house) is the object, and the verb کشیدم (drew) comes at the very end. This is different from English, which uses a Subject-Verb-Object order.