Yes, in this case, the "to" has been omitted by accident. Actually, each case must be learned, because some verbs are followed by the base form with no “to” (they let him do it), some are followed by the gerund (they tolerated his doing it), and yes, some by the infinitive with “to.” I think the dropping of the “to” in this case has to do with the spoken language, where the /u/ in “to” is reduced to schwa, and the preceding -ed combines with the “t” of “to.”

Yes, the "to" is necessary. That one stood out to me when I read it, before I read about your question over the sentence. When you have two verbs next to each other like that, the second verb will usually be in the infinitive form (infinive = "to + verb").