Learn Croatian: Imam nekoliko pitanja. - I have several questions.

Because of the word nekoliko (several). Words that express quantity require the noun that follows them to be in the genitive case. Since there is more than one question, we use the genitive plural form.

The dictionary form is pitanje. Does it have the exact same ending for both the normal plural and the genitive plural?

Yes! For neuter nouns like pitanje, the regular plural (nominative) is pitanja, and the genitive plural is also spelled pitanja. In spoken Croatian, the final -a in the genitive plural is drawn out and pronounced slightly longer to tell them apart, but in writing, they look identical.

Are there other words like nekoliko that force the noun into the genitive plural?

Yes, this is a very common rule! Other words of quantity like puno (a lot), malo (a few), and numbers from five upwards all work exactly the same way. When they are used to count something, they must be followed by the genitive plural.

I don't see the word for "I" (ja) in this sentence. Is it okay to just start with imam?

Absolutely. In Croatian, the verb ending already tells us who is doing the action. Imam means "I have." You only need to add ja if you want to emphasize that you have questions, as opposed to someone else.