Learn French: Ce journal est gratuit. - This newspaper is free.

No, because gratuit specifically means "free of charge." The word libre means "free" in the sense of having freedom, being unoccupied, or not being restricted (like a free seat or a free country).

Does journal mean "journal" (like a personal diary) or "newspaper"?

In everyday French, un journal most commonly refers to a "newspaper." If you want to talk about a personal diary, you would normally use the specific phrase un journal intime.

Why do we use ce for "this" instead of cet?

Because journal is a masculine singular noun that starts with a consonant. You only use cet for masculine singular nouns that start with a vowel or a silent H (for example, cet arbre or cet homme).

How would this exact sentence change if we were talking about multiple free newspapers?

It would become Ces journaux sont gratuits. The word journal has an irregular plural form ending in -aux, the plural of ce is ces, and you must add an -s to gratuit so the adjective matches the plural noun.