Learn Finnish: Opiskelija lakkaa vihdoin valittamasta. - The student finally stops complaining.

The verb lakata (to stop or cease) requires the action you are stopping to be in the third infinitive elative form, which ends in -masta or -mästä. You can think of it as stepping away from an action, which is why it uses an ending that normally means from or out of.

How do we build the form valittamasta from the dictionary verb?

The dictionary form is valittaa (to complain). To make the third infinitive, you take the strong stem valitta-, add the third infinitive marker -ma-, and then add the elative ending -sta. This gives you valittamasta.

Why does the verb lakkaa (stops) have two Ks when the dictionary form only has one?

The dictionary form is lakata (to stop), which is a verb type 4 word. These verbs have reverse consonant gradation. This means the dictionary form has the weak consonant grade (one K), but the conjugated stem becomes strong and gets two Ks, making the stem lakkaa-.

Could I use the verb loppua instead of lakata here?

Not in this sentence. You use lakata when a person or subject stops doing a specific action. You use loppua when a thing or event ends on its own, for example Kurssi loppuu (The course ends).