Do the guest schools spend an entire academic year in the host school during the Triwizard Tournament?

Certainly the guest school's students stay at the host school for the entire year - there are plenty of mentions of the Durmstrang and Beauxbatons students being around throughout Goblet of Fire at numerous times, aside from the tasks themselves and the Yule Ball.

There's nothing concrete in the text, but I would imagine the guest students attend the same classes as their hosts. There weren't a huge number of them so they wouldn't affect class sizes too much, and remember they were all several years older than Harry which is why we'd never see them in lessons.

It's unlikely that the students' respective schools would allow them to spend a whole year effectively on holiday!

The delegation of students from the foreign schools all stay at Hogwarts for the entire Tri-wizard tournament.

The most likely explanation is that the students continue their studies together in their respective residences on Hogwarts campus.

The book gives no reference of any shared classes between Hogwarts students and others, and mentions multiple times how the other schools students all leave their respective housing together, trudge up to Hogwarts for breakfast, lunch, dinner, and then trudge back down to their carriage/boat.

In fact taking lessons at Hogwarts would be unusual given the competition between the 3 schools.

‘There’s traditionally been a lot of rivalry between all the magic schools. Durmstrang and Beauxbatons like to conceal their whereabouts so nobody can steal their secrets,’ said Hermione matter-of-factly.

Here we see how strongly Karkaroff at least wants his schools secrets kept, most likely due to the emphasis on the dark arts the school is rumored to have.

‘Now, now, Viktor!’ said Karkaroff, with a laugh that didn’t reach his cold eyes. ‘Don’t go giving away anything else, now, or your charming friend will know exactly where to find us!’

Dumbledore smiled, his eyes twinkling. ‘Igor, all this secrecy ... one would almost think you didn’t want visitors.’

‘Well, Dumbledore,’ said Karkaroff, displaying his yellowing teeth to their fullest extent, ‘we are all protective of our private domains, are we not? Do we not jealously guard the halls of learning that have been entrusted to us? Are we not right to be proud that we alone know our school’s secrets, and right to protect them?’

‘Oh, I would never dream of assuming I know all Hogwarts’ secrets, Igor,’ said Dumbledore amicably.

Allowing your students access to other teachers and ideas is dangerous when your own teaching and ideas may be frowned upon by others such as his students leaking how much time they spend on the dark arts. Id expect the Durmstrang students to be isolated as much as possible, other then Krum, who was allowed access to Hogwarts extensive library to increase his knowledge for the tasks ahead.

Viktor Krum was in the library an awful lot, too, and Harry wondered what he was up to. Was he studying, or was he look- ing for things to help him through the first task?

It's never addressed in the book whether the foreign students attended classes at Hogwarts or if their respective headmaster/headmistress taught them, so we can only speculate.

In the one hand, taking classes at Hogwarts can be problematic:

• There is the language barrier: It seems not all of the foreign students speak English (at least not on a level required for advanced Newt-classes)

• The curriculum and ministry requirements may be different: Durmstrang, for example, teaches the dark arts, not just DADA. An ambitious Durmstrang student who wants to specialise in the dark arts won't be able to take the relevant class at Hogwarts.

• A headmaster like Karakaroff may object to expose his students to the too liberal, Hogwarts-style education or even might insist on teaching the champion (and the rest) himself in order to prepare him for the tournament fearing that the Hogwarts teachers would sabotage his students and hide information about spells that might prove useful in the tournament. After all they didn't even agree to lodge in Hogwarts and I don't think the reason was the castle being overcrowded. There's plenty of free space there - after all, in Harry's first year Dumbledore closed off an entire floor Fluffy was in.

On the other hand it's a good opportunity to learn new things and to see different methods implemented, and it might well worth to retake some classes at home.

The ministries of the respective countries probably made arrangements beforehand concerning the lodging, use of the library, meals and classes, but we have no information about the last bit.

"Harry, whose attention had been focused completely upon Madame Maxime, now noticed that about a dozen boys and girls, all, by the look of them, in their late teens, had emerged from the carriage and were now standing behind Madame Maxime."

^This passage says that there's around 12 students (maybe that huge carriage was a little over-kill for a relatively small amount of kids), all in grade 6 or 7. And I assume it would be pretty similar for the Durmstrangs. Headmasters of any kind of school, muggle-y or magic, are usually very well-versed in school curricula, I mean Dumbledore himself apparently OWLed 'Outstandingly' in all his subjects like Hermione did. I suppose the Heads of Beauxbaton's and Durmstrang's just slogged through and taught their lot home-school style every day.

As well as the circumstantial evidence, Dumbledore tells us directly that they are.

"The delegations from Beauxbatons and Durmstrang will be arriving in October, and remaining with us for the greater part of this year. I know that you will all extend every courtesy to our foreign guests while they are here..."
The Triwizard Tournament, GoF

In terms of lessons, we're never told. We know that

..."there were barely twenty additional students.."
The Goblet of Fire, GoF

so they could probably integrate into NEWT classes without causing too much difficulty (assuming they don't all take exactly the same subjects), or be taught by their own headteachers. Harry wouldn't see either of these scenarios.

We do know that they sleep in the carriage and boat, rather than the castle (numerous quotations about them going to/from the grounds) which might indicate some level of separation, and make being taught separately more likely, but there's no canon evidence for that.