Learn Dutch: Verder moet je het luchtbed opblazen. - Furthermore, you have to inflate the air mattress.

No. Only the actual pronoun er (and sometimes locatives like hier, daar, or waar) interact with 'er' deletion rules. The letters 'er' inside a regular adverb like verder have no grammatical effect on whether you can use the pronoun er elsewhere. For example, you can safely say Verder is er niets (Furthermore, there is nothing).

Why does the sentence use the word order moet je instead of je moet?

Because the sentence starts with the adverb Verder, the conjugated verb moet must take the second position to satisfy the V2 (verb-second) rule. This pushes the subject je to the third position, causing inversion.

Opblazen is a separable verb. Why isn't it split in this sentence?

Separable verbs only split when they are the conjugated verb in a main clause. Here, the modal verb moet is the conjugated verb, which forces the main verb opblazen to the very end of the sentence in its full, unsplit infinitive form.

If I wanted to replace het luchtbed with a pronoun, would I use er?