Learn Spanish (Latin America): Incluso con el pinchazo, el ciclista pedalea rápido. - Even with the puncture, the cyclist pedals fast.

Yes, both incluso and hasta can mean even to introduce a surprising or extreme element. In this sentence, they are completely interchangeable. You could say Hasta con el pinchazo... and the meaning would be exactly the same.

How would I use ni siquiera in a sentence like this?

To say not even, you would use ni siquiera. Because it introduces a negative idea, the main action usually changes to match. For example, you might say Ni siquiera con el pinchazo dejó de pedalear rápido (Not even with the puncture did he stop pedaling fast).

Why is the article masculine (el) when ciclista ends in -a?

Nouns ending in -ista that refer to people (like professions or hobbies) do not change their endings. Instead, the article tells you the gender. Here, el ciclista refers to a male cyclist or cyclists in general, while la ciclista would be a female cyclist.

Why does the sentence use rápido instead of the adverb rápidamente?

In everyday Spanish, the adjective rápido is frequently used as an adverb meaning fast or quickly without adding -mente. While rápidamente is grammatically correct, using just rápido is a very common flat adverb that sounds much more natural and conversational in this context.