Learn Swedish: Svindlaren låtsas att han är hur rik som helst. - The swindler pretends that he is incredibly rich.

Låtsas is a deponent verb, which means it ends in -s (like a passive verb) but has an active meaning. For this specific verb, the infinitive and the present tense forms are exactly the same: låtsas.

Could we use an infinitive here instead of the att-clause?

Yes! Because the swindler is pretending something about himself, you could use a bare infinitive: Svindlaren låtsas vara hur rik som helst. You only strictly need an att-clause when the person pretending is different from the subject of the pretense (for example, if he were pretending that she was rich).

How does hur rik som helst translate to "incredibly rich"?

The structure hur [adjective] som helst is a very common Swedish idiom that means "incredibly [adjective]" or "as [adjective] as can be". Literally it translates closer to "how rich as whatsoever". You can use this pattern with almost any adjective, like hur bra som helst (incredibly good).

Is it possible to drop the word att in this sentence?

Yes, just like the word "that" in English, the subjunction att is frequently dropped in casual, spoken Swedish. You will often hear Svindlaren låtsas han är..., though keeping att is the safest choice for written Swedish.