Does your country have a “national poet” and if so, are his or her poems studied in public schools?

We do and Canada's finest is Leonard Cohen. Cheers! 

Yes and it is supposedly a one-year term but the last one that I know of and could find was appointed in 2019, Joy Harjo.  I find no explanation for this but I assume a political one.  Sad, isn't it.....

Andrew Barton "Banjo" Paterson is probably our best known poet,  he authored our best known folk song and unofficial anthem, Waltzing Matilda, and The Man From Snowy River,  among others. He's not considered to be the "national" poet, though. Other notables include Henry Lawson, Dorothea Mackellar, CJ Dennis, Father John O'Brien (yes, he was a Catholic priest) and indigenous poet "Kath Walker" (Oodgeroo Noonuccal).
In my senior year in school I was required to study the poetry of John Foulcher, a close friend of my English teacher but that didn't improve his poetry which I still consider to be sadomasochistic pseudo-intellectual masturbation.

The writers who would be considered "national poets" from my country are José Hernández, Jorge Luis Borges and Leopoldo Lugones. I studied their poetry and some of their other writings in school. I remember that I was assigned to write reports about Jorge Luis Borges's work when I was in high school. He was the writer that we studied the most because he was the most contemporary writer. I remember staying up late and being sleepy from writing those reports.

 

  It’s most likely the person named Poet Laureate once a year, but I know absolutely nothing about poetry, I am not a patron of it. That’s despite the fact that my mother is a lifelong poet, so I grew up surrounded by it. 

  I was originally going to stop there, but my natural curiosity, combined with my aching desire for accuracy, got the best of me. I had to look up exactly what is entailed in this laureate jazz, so here’s a very condensed version of what I found out:America stole the idea from the British in 1937, the name of the honor was changed in 1985, and there is a separate awarding of it for children’s poetry.
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In Scotland it's Rabbie Burns. In my day it was studied in schools. Don't know about nowadays. The Old Scots vernacular is probably too much like hard work.  

Here's your lecture on Milton: