Learn Italian: Si tratta di un problema molto serio. - It is a matter of a very serious problem.

No. While it is an impersonal use of si, the phrase si tratta di is a fixed expression meaning 'it is a matter of'. Because it doesn't use the verb essere (to be) to describe a general group of people, the rule you are currently learning about using plural adjectives (like si è stanchi) doesn't apply here.

Why is it un problema molto serio instead of una problema molto seria?

Even though problema ends in -a, it is actually a masculine noun. Many Italian words of Greek origin ending in -ma (like il problema, il sistema, il clima) are masculine. Therefore, it takes the masculine article un and the masculine adjective serio.

If we were dealing with multiple serious problems, would the verb change to si trattano di?

No, it stays si tratta di problemi molto seri. The phrase trattarsi di is strictly impersonal and only ever conjugates in the third-person singular (tratta), regardless of whether the noun that follows it is singular or plural.

Could I say Questo libro si tratta di un problema serio to mean 'This book is about a serious problem'?