Learn Italian: Dobbiamo selezionare gli scatoloni e metterli via ordinatamente. - We must select the boxes and put them away neatly.

In Italian phrasal verbs formed with an adverb like via, su, or giù, object pronouns must attach directly to the verb itself, never to the adverb. Since mettere is in the infinitive here, we drop the final -e and attach li to make metterli, keeping the adverb via as a separate word.

The normal word for box is scatola (feminine). Why does this sentence use gli scatoloni (masculine plural)?

This uses the augmentative suffix -one to indicate a large box, such as a moving box or a large cardboard box. When this suffix is added to a feminine noun ending in -a (like la scatola), the resulting 'big' noun almost always changes gender and becomes masculine: lo scatolone (plural: gli scatoloni).

Could we use pronoun climbing here and move li to the very beginning, before dobbiamo?

Not in this specific sentence. While you can usually move a pronoun before a modal verb like dovere, here li only serves as the object for the second verb (mettere). The first verb (selezionare) already has its explicit object (gli scatoloni). Moving li to the start would tangle the syntax of the two coordinated verbs.