Learn Swedish: Du kommer att vilja dra armstödet bakåt. - You will want to pull the armrest backward.

bakåt describes a movement in a backward direction or trajectory, which perfectly fits the continuous action of pulling. bak usually refers to a static location (at the back), and tillbaka means back in the sense of returning to an original place or state (like giving an object back).

Why is there no att before dra in the phrase kommer att vilja dra?

In Swedish, when you stack verbs, the infinitive marker att is generally only used before the first infinitive, and only if required by the first auxiliary. Since vilja (want) acts as a modal auxiliary for the next verb, dra (pull) simply takes the bare infinitive.

Would a Swede actually say the word att in this sentence during a normal conversation?

Probably not. In spoken and casual written Swedish, the att in the future tense marker kommer att is very often dropped, so you will frequently hear Du kommer vilja dra instead.

How can we tell the gender of the noun stöd from the compound armstödet?

Because the compound noun ends in the definite suffix -et, we know that the root noun is an ett-word: ett stöd (a support or rest). In Swedish compound nouns, the very last word always dictates the gender of the entire word.