Learn Hardcore Finnish: Millainen tämä kahvila on?

Millainen tämä kahvila on? → asking for a description (cozy, big, noisy…)
If you asked Mikä tämä kahvila on?, you would be asking for identity (Which café is this? Starbucks or something else?)

So millainen ≈ what kind of, whereas mikä ≈ what / which (one).

Why does the word order look like Millainen tämä kahvila on? and not more like English What kind this café is?

In Finnish, a question word (here millainen) normally comes first in the sentence.

The basic structure here is:

Millainen (question word)
tämä kahvila (subject phrase)
on (verb olla, “to be”)

So the most neutral order is Millainen tämä kahvila on?.
You can also say Millainen kahvila tämä on?; both are correct and natural, with only a slight difference in emphasis (see below).

Is there any difference in meaning between Millainen tämä kahvila on? and Millainen kahvila tämä on?

Both can be translated as What is this café like? and are usually interchangeable.

Millainen tämä kahvila on? – slightly more emphasis on this specific café.
Millainen kahvila tämä on? – slightly more emphasis on the type / kind of café this is.

In everyday conversation, most learners can treat them as meaning the same thing; both sound natural.

Why is the verb on at the end? Could you leave it out?

On is the 3rd person singular of olla (to be) and it must be there in standard written Finnish.
The normal word order with the verb olla in questions like this is to place on near the end:

Tämä kahvila on kiva. (statement)
Onko tämä kahvila kiva? (yes/no-question)
Millainen tämä kahvila on? (wh-question)

In some colloquial spoken Finnish, people may drop on in similar sentences, but for learning and for correct Finnish you should always include on here.

What role does tämä kahvila play in the sentence? Is it the subject?

Yes. Tämä kahvila is the subject.

tämä = this
kahvila = café

The underlying structure is like a normal X is Y sentence:

Tämä kahvila (subject) on (verb) millainen? (complement being asked about)

The question word millainen stands for the part of the sentence you don’t yet know (the description), so you move it to the front to form a question.

Why is there no word like a or the before kahvila?

Finnish does not have articles like English a / an / the.

kahvila can mean a café, the café, or just café, depending on context.
tämä kahvila means literally this café, but that can function as either this café or this particular café in English.

So the sentence needs no extra word for a / the; the demonstrative tämä already gives enough information.

Why is it tämä kahvila and not something like tässä kahvilassa?

Tämä kahvila is in the nominative case and acts as the subject: it simply points to this café as an object in the world.

Tässä kahvilassa is inessive case (in this café) and would answer a different kind of question, for example:

Millainen tässä kahvilassa on tunnelma? – What is the atmosphere like in this café?

So tämä kahvila = this café (as a thing),
tässä kahvilassa = in this café (location).

What is the difference between millainen and minkälainen?

Functionally, they mean the same thing: what kind of / what…like.

millainen tämä kahvila on?
minkälainen tämä kahvila on?

Both are correct and widely used.
There are minor stylistic and regional preferences (many people say minkälainen in speech), but as a learner you can treat them as synonyms in sentences like this.

How would you make a similar question in the plural, like What are these cafés like?

For plural, you change both the demonstrative and the verb, and usually also millainen:

Millaisia nämä kahvilat ovat? – What are these cafés like?

Notes:

nämä = plural of tämä (these)
ovat = plural of on (are)
millaisia is the plural partitive of millainen, commonly used when asking about a group’s qualities.

You might also hear Millaisia nämä kahvilat on? in colloquial speech, but in standard language it should be ovat.

How would you typically answer Millainen tämä kahvila on?

You answer with a description using an adjective or adjective phrase:

Tämä kahvila on pieni. – This café is small.
Tämä kahvila on viihtyisä ja hiljainen. – This café is cozy and quiet.
Tämä kahvila on todella suosittu. – This café is really popular.

The pattern is:

Tämä kahvila (subject) + on (verb) + [adjective/description].

Where does the form kahvila come from, and what does -la mean?

Kahvila is built from:

kahvi = coffee
-la / -lä = a common suffix meaning place associated with X

So kahvila literally means a place of coffee, i.e. a café or coffee shop.
You see the same pattern in many words:

kirjasto (different suffix, but same idea): kirja (book) → kirjasto (library)
koulu (school) + -la → koululla (at the school, location form)

In kahvila, the -la is part of the basic noun, so you just learn kahvila as café.