Yes, in fact most of the creatures he has in class aren’t monsters.
Hagrid does show his class the Blast-Ended Skrewts, but actually the majority of the creatures he brings to teach his class about (as opposed to his own personal pets) aren’t monsters. The Thestrals were a bit creepy, perhaps, but overall he doesn’t usually actually teach about or bring monsters to class - the only really ‘monstrous’ things he tried to teach about were the Skrewts.
Nifflers (XXX - Competent wizard should cope):
Hagrid has his class use Nifflers to hunt for leprechaun gold.
“These’re Nifflers,’ said Hagrid, when the class had gathered around. ‘Yeh find ’em down mines mostly. They like sparkly stuff … there yeh go, look.”
- Harry Potter and the Goblet of Fire, Chapter 28 (The Madness of Mr. Crouch)
They’re actually described as being gentle and affectionate.
“Though the Niffler is gentle and even affectionate, it can be destructive to belongings and should never be kept in a house.”
- Fantastic Beasts and Where to Find Them
They’re cute, fluffy, and affectionate - definitely not monsters.
Salamanders (XXX - Competent wizard should cope):
Hagrid brings salamanders for his class on a cold day.
“Lessons started again next day. The last thing anyone felt like doing was spending two hours in the grounds on a raw January morning, but Hagrid had provided a bonfire full of salamanders for their enjoyment, and they spent an unusually good lesson collecting dry wood and leaves to keep the fire blazing, while the flame-loving lizards scampered up and down the crumbling, white-hot logs.”
- Harry Potter and the Prisoner of Azkaban, Chapter 12 (The Patronus)
There’s nothing particularly dangerous or monstrous about them.
“The salamander is a small fire-dwelling lizard that feeds on flame. Brilliant white, it appears blue or scarlet depending upon the heat of the fire in which it makes its appearance.”
- Fantastic Beasts and Where to Find Them
They don’t seem dangerous, and Harry considered it a good lesson, so they probably weren’t.
Flobberworms (X - Boring):
Hagrid really only taught about Flobberworms because he was worried after Buckbeak bit Draco, but he’d still have to know about them to be able to teach about them.
“Nobody really liked Care of Magical Creatures, which, after the action-packed first class, had become extremely dull. Hagrid seemed to have lost his confidence. They were now spending lesson after lesson learning how to look after Flobberworms, which had to be some of the most boring creatures in existence.”
- Harry Potter and the Prisoner of Azkaban, Chapter 8 (Flight of the Fat Lady)
They’re incredibly boring, which is why no one enjoyed the lessons.
“The Flobberworm lives in damp ditches. A thick brown worm reaching up to ten inches in length, the Flobberworm moves very little.”
- Fantastic Beasts and Where to Find Them
They were no one’s first choice, but Hagrid still knew how to take care of them.
Hippogriffs (XXX - Competent wizard should cope):
Hippogriffs also weren’t monsters, despite the lesson ending badly due to a student’s misbehavior.
“Well done, Harry!’ said Hagrid, ecstatic. ‘Right – yeh can touch him! Pat his beak, go on!’
Feeling that a better reward would have been to back away, Harry moved slowly towards the Hippogriff and reached out towards him. He patted the beak several times and the Hippogriff closed his eyes lazily, as though enjoying it.”
- Harry Potter and the Prisoner of Azkaban, Chapter 6 (Talons and Tea Leaves)
Hagrid gave his class clear safety instructions on how to approach them, which Draco didn’t follow.
“Yeh always wait fer the Hippogriff ter make the firs’ move,’ Hagrid continued. ‘It’s polite, see? Yeh walk towards him, and yeh bow, an’ yeh wait. If he bows back, yeh’re allowed ter touch him. If he doesn’ bow, then get away from him sharpish, ’cause those talons hurt.”
- Harry Potter and the Prisoner of Azkaban, Chapter 6 (Talons and Tea Leaves)
Draco not listening doesn’t make Hippogriffs monsters or Hagrid negligent in this case.
Unicorns (XXXX - but only because they require respect when handling)
Though he’s just continuing Professor Grubbly-Plank’s lessons, Hagrid is knowledgeable about unicorns and is able to continue her lessons on them.
“Easier ter spot than the adults,’ Hagrid told the class. ‘They turn silver when they’re abou’ two years old, an’ they grow horns at aroun’ four. Don’ go pure white ’til they’re full-grown, round about seven. They’re a bit more trustin’ when they’re babies … don’ mind boys so much … c’mon, move in a bit, yeh can pat ’em if yeh want … give ’em a few o’ these sugar lumps …”
- Harry Potter and the Goblet of Fire, Chapter 26 (The Second Task)
They’re one of the highest Ministry classifications he’s taught about, but not dangerous.