Learn Italian: La gente ha subito notato il loro nipotino. - People immediately noticed their little grandson.

Normally, yes! We drop the definite article for singular family members (like mio fratello). However, the possessive loro is an exception to this rule. It always keeps the definite article, no matter what noun follows it.

If it were "my little grandson" instead of "their little grandson", would it still be il mio nipotino or just mio nipotino?

It would still be il mio nipotino, which brings us to another exception! When a family member noun is modified by a suffix—like the diminutive -ino in nipotino (little grandson)—it keeps the definite article. So it's mio nipote (no article) but il mio nipotino (with the article).

Why is the verb ha notato in the singular when la gente means "people" (plural)?

In Italian, la gente is a singular feminine collective noun. Think of it like the English word "the crowd." Even though it refers to many individuals, grammatically it is just one group, so it always takes a third-person singular verb (ha), never plural.

Is it mandatory to put subito between the two parts of the verb (ha and notato)?