Learn Italian: Qui è severamente vietato fumare. - Here it is strictly forbidden to smoke.

The subject is actually the infinitive verb fumare (to smoke). When an infinitive acts as the subject of a sentence, any adjectives or past participles describing it, like vietato, must take the default masculine singular ending -o.

How could we include chiunque to emphasize that this applies to absolutely everyone?

You could say Qui è severamente vietato a chiunque fumare (Here it is strictly forbidden for anyone to smoke). Even though the original sentence already implies a universal rule, adding a chiunque strongly emphasizes that there are zero exceptions for anybody.

Why is the adverb severamente placed between è and vietato?

In Italian, adverbs of manner (which often end in -mente) frequently sit right between the auxiliary verb (è) and the past participle (vietato) for a natural flow. You might also see the adverb placed after the participle, but this middle position is very common and adds emphasis to the strictness of the rule.

Could I use the gerund fumando instead of the infinitive fumare here?

No, you must use the infinitive fumare. In English we can say 'smoking is forbidden', which uses an -ing word as a noun. In Italian, you cannot use the gerund (fumando) as a noun; you must always use the infinitive when a verb is acting as the subject of the sentence.