Learn Portuguese (Portugal): Ontem, falei com o professor. - Yesterday, I spoke with the teacher m.

The base verb is falar (to speak). To talk about a completed action in the past, we drop the -ar and add -ei for the "I" form. So, falo means "I speak" (present), but falei means "I spoke" (past).

Why isn't the word eu (I) included before falei?

In Portuguese, the verb ending gives away who is doing the action. Since the -ei ending on a past-tense -ar verb is only ever used for "I", adding the word eu is optional. Leaving it out sounds very natural to native speakers.

Does com o ever combine into a single word?

No. You might remember from earlier lessons that words like de (of/from) and em (in/on) merge with o to make do and no. However, com (with) does not shrink or merge with the word for "the". It stays as two separate words: com o.