Learn Finnish: Jos lauta on pitkä, leikkaan sen sahalla. - If the board is long, I cut it with a saw.

In Finnish, you do not use a separate preposition for "with" when talking about the tool or instrument you use to do something. Instead, you add the Adessive ending -lla or -llä directly to the noun. So, saha (saw) becomes sahalla (with a saw).

Why is the word for "it" in the form sen instead of sitä?

Here sen is used as a total object. You use it because cutting the board is viewed as a complete action done to one specific, whole item. If the action were ongoing ("I am cutting it") or if you were cutting an undefined amount of a substance, you would use the partitive form sitä.

What is the dictionary form of leikkaan, and why does it have a double k here?

The dictionary form is leikata (to cut), which is a Type 4 verb. Type 4 verbs undergo reverse consonant gradation when conjugated. This means the weak single k in the basic form strengthens to a double kk in conjugated forms like leikkaan (I cut).

In the second clause, do I need to include a word for "then"?

No, it is not strictly required. While you will often hear Finnish speakers use niin (then) to start the main clause after a jos (if) clause, it is perfectly natural and correct to just use a comma, exactly as this sentence does.