Learn Hardcore Finnish: Laboratoriossa teemme ryhmätöitä kahden muun opiskelijan kanssa. - In the laboratory we do group work with two other students.

This is also correct. The difference is mostly in emphasis and style, not grammar. Starting with the place (laboratoriossa) is very common in Finnish when you want to set the scene.

What form is teemme, and why is there no word for we?

Teemme is the 1st person plural (we) form of the verb tehdä (to do, to make).

infinitive: tehdä = to do
minä teen = I do
me teemme = we do

Finnish usually drops subject pronouns (like minä, me) because the verb ending already shows who the subject is. So:

Me teemme ryhmätöitä...
Teemme ryhmätöitä...

Both mean we do group work..., but the version without me is more neutral and typical in Finnish.

What exactly is ryhmätöitä and why does it look like that?

Ryhmätöitä comes from the noun ryhmätyö:

ryhmä = group
työ = work
ryhmätyö = group work (often a specific task, project, or assignment)

The form ryhmätöitä is:

plural
partitive case

So ryhmätöitä literally means (some) group works / group work tasks.

Finnish uses the partitive plural here because we are talking about:

work in general or an indefinite amount of group work
an activity rather than a specific, countable number of tasks

English just says group work, but Finnish marks this with the partitive plural ryhmätöitä.

Why not just say ryhmätyöt instead of ryhmätöitä?

Ryhmätyöt (nominative plural) would sound like:

specific, countable group work tasks (like “the group assignments”)
a more complete, delimited set of tasks

Ryhmätöitä (partitive plural) feels more like:

some group work
group work as an ongoing activity, not a fixed, completed set

In this sentence, ryhmätöitä matches the idea of doing group work in general, not necessarily finishing a specific list of tasks.

What does kahden mean, and why not just kaksi?

Both come from the number kaksi (two):

kaksi is the basic form (nominative)
kahden is the genitive form of kaksi

You use kahden when it has to be in genitive, for example:

kahden opiskelijan kanssa = with two students
(literally: with of-two students)

The word kanssa (with) requires its complement to be in the genitive case, so kaksi must change to kahden here.

What is muun doing in kahden muun opiskelijan kanssa?

Muu means other.

basic form: muu = other
genitive singular: muun = of (an) other

In Finnish, adjectives must agree in case and number with the noun they modify. Here:

kahden = genitive of kaksi (two)
muun = genitive of muu (other)
opiskelijan = genitive of opiskelija (student)

So kahden muun opiskelijan = of two other students.
This whole phrase goes with kanssa, which requires genitive.

Why is opiskelijan singular when the meaning is two students?

This is a common pattern with numbers in Finnish.

With kaksi (two) and kahden (of two), the noun often appears in a singular-looking form, but grammatically it behaves like a special form connected to numbers.

kaksi opiskelijaa = two students (nominative context)
kahden opiskelijan kanssa = with two students (genitive context)

So even though opiskelijan looks like singular genitive (of a student), in this structure it goes together with kahden and really means of two students. This is normal Finnish number–noun behaviour.

What does kanssa mean, and why does it come after the noun phrase?

Kanssa means with.

It is a postposition, not a preposition. That means it usually comes after the word or phrase it relates to, and that word/phrase is in the genitive case:

ystävän kanssa = with a friend
opettajan kanssa = with the teacher
kahden muun opiskelijan kanssa = with two other students

So the structure is:
[genitive phrase] + kanssa = with [that phrase].

Can I move kahden muun opiskelijan kanssa to the beginning or middle of the sentence?

Yes, Finnish allows flexible word order, as long as the case endings stay correct. For example:

Laboratoriossa teemme ryhmätöitä kahden muun opiskelijan kanssa.
Laboratoriossa kahden muun opiskelijan kanssa teemme ryhmätöitä.
Kahden muun opiskelijan kanssa teemme laboratoriossa ryhmätöitä.
Teemme ryhmätöitä kahden muun opiskelijan kanssa laboratoriossa.

All are grammatically correct. What changes is:

what is emphasized (place, company, activity)
what sounds most natural in context

The original version is very natural and neutral.

Is ryhmätöitä one word or two? Why is it written together?

Ryhmätöitä is one word because ryhmätyö is a compound noun:

ryhmä
- työ → ryhmätyö (group work / group task)
partitive plural: ryhmätöitä

In Finnish, when two nouns form a single concept, they are usually written together as one word. You then add case endings to the whole compound.

Could I say teemme töitä ryhmässä instead of teemme ryhmätöitä?

You could say teemme töitä ryhmässä, but it is not exactly the same:

teemme ryhmätöitä
– we do group work / group assignments (strongly suggests school-type group tasks)
teemme töitä ryhmässä
– we work in a group (describes the situation: working in a group, but not necessarily formal group assignments)

Both are understandable, but ryhmätöitä fits better for typical student group work in a lab setting.