Learn Hardcore Swedish: Jag sitter på trappan och dricker kaffe. - I am sitting on the stairs and drinking coffee.

Jag sitter på trappan = literally I sit on the stairs, but it means I am sitting on the stairs right now.
Jag är på trappan would mean I am on the stairs (located there), but it does not clearly say that you are sitting; you could be standing or doing something else.

So:

To say what you are doing with your body right now: use sitter / står / ligger.
To say only where you are (location): use är.

That is why Jag sitter på trappan is the normal, natural choice here.

Is sitter a special continuous form like English am sitting?

No. Swedish does not have a special continuous tense like English -ing forms.

sitter is just the simple present form of sitta (to sit).
The same form is used for:
- right now: Jag sitter på trappan. = I am sitting on the stairs (now).
- habitually: Jag sitter ofta på trappan. = I often sit on the stairs.

Swedish normally uses the simple present for both I sit and I am sitting.
If you really want to stress that it is happening right now, you can add nu (now), but it is not required:

Jag sitter på trappan nu. = I am sitting on the stairs now.

Why is it på trappan and not something like i trappan or vid trappan?

The preposition på is used for being on top of a surface, like:

på stolen = on the chair
på bordet = on the table
på trappan = on the stair(s) / on the step

Alternatives:

i trappan
Literally in the stairs; this sounds wrong for ordinary sitting. You normally don’t sit in the stairs.
vid trappan
Means by the stairs, next to the stairs, not actually on them.

So if you are literally sitting on a step, på trappan is the natural preposition.

Why does trappan have -an at the end? What is the difference between trappa and trappan?

Trappa is a common gender noun (an en-word):

en trappa = a stair / a staircase
trappan = the stair / the staircase

So:

Jag sitter på en trappa = I am sitting on a stair (some stair, not specified).
Jag sitter på trappan = I am sitting on the stair / the stairs (a particular one we both know or can see).

In everyday use, trappan is often translated as the stairs, not just the stair, because Swedish tends to use trappa where English uses stairs or steps.

In the sentence, trappan suggests both speaker and listener know which stairs are meant (for example the stairs outside the house).

Why is it singular trappan and not a plural form when English often says on the stairs?

Swedish uses trappa / trappan in a slightly different way than English uses stairs:

en trappa can refer to a set of steps (a staircase), not just one single step.
trappan often corresponds to English the stairs.

If you really wanted to focus on multiple separate steps in Swedish, you could say:

steg = steps (as in individual steps you take).

But for sitting on the stairs, Swedes usually just say på trappan, even though English uses a plural form (on the stairs).

Why is there no article before kaffe? Why not something like dricker ett kaffe?

In this sentence, kaffe is treated as a mass noun (an uncountable substance), like English coffee, water, tea:

Jag dricker kaffe. = I am drinking coffee.
(some coffee, not specified, uncountable)

You can use articles with kaffe, but the meaning changes:

Jag dricker ett kaffe.
Colloquial, usually means I’m having a coffee (one cup of coffee, often in a café context).
Jag dricker kaffet.
Means I am drinking the coffee (the specific coffee that has already been mentioned or is known).

In your sentence, the neutral, general idea drinking coffee is wanted, so no article is used: dricker kaffe.

Is the verb form dricker also just simple present? Does it mean both drink and am drinking?

Yes. Dricker is the simple present form of dricka (to drink).

It covers both English meanings:

Jag dricker kaffe.
- I drink coffee (habitually).
- I am drinking coffee (right now).

Which one is intended depends on the context. In the sentence Jag sitter på trappan och dricker kaffe, the context (together with sitter) makes it clear that it is happening right now.

Why is the subject jag not repeated before dricker? Could I say Jag sitter på trappan och jag dricker kaffe?

Both are grammatically correct:

Jag sitter på trappan och dricker kaffe.
Jag sitter på trappan och jag dricker kaffe.

In Swedish, when the subject is the same for both verbs, it is normal to drop it in the second clause:

Jag sitter på trappan och dricker kaffe.

Adding jag again is possible, but it often sounds heavier or more emphatic, like you are stressing the second part:

Jag sitter på trappan och jag dricker kaffe.
(could sound like you are listing things you do, with emphasis)

Could the sentence be Jag sitter på trappan och dricker without kaffe?

Yes, grammatically that is fine:

Jag sitter på trappan och dricker. = I’m sitting on the stairs and drinking.

However, without an object it is less specific. In many contexts, listeners will guess you mean drinking something alcoholic if you just say dricker with no object, especially in casual speech.

If you want to clearly mean coffee, it is better to say dricker kaffe, as in the original sentence.

Can I change the word order to something like På trappan sitter jag och dricker kaffe?

Yes. That is a perfectly good sentence, just with a slightly different emphasis.

Swedish main clauses follow the V2 rule (the verb is in second position):

Jag sitter på trappan och dricker kaffe.
Order: Subject (Jag) – Verb (sitter) – Rest.

If you move a different element to the beginning, the finite verb still has to be second:

På trappan sitter jag och dricker kaffe.
Order: Adverbial (På trappan) – Verb (sitter) – Subject (jag) – Rest.

This version puts a bit more focus on the location (på trappan).
But both versions are correct and mean practically the same thing in most contexts.

What exactly does och do here? Is it joining two sentences or just two verbs?

In Jag sitter på trappan och dricker kaffe, och (and) is joining two verb phrases that share the same subject jag:

[jag] sitter på trappan
[jag] dricker kaffe

So och here links:

sitter på trappan and
dricker kaffe

You could also make them into two separate sentences:

Jag sitter på trappan. Jag dricker kaffe.

But Swedish (like English) often prefers to coordinate related actions with och when they happen at the same time.

How is dricker pronounced, and is there anything tricky about it for English speakers?

Dricker is pronounced roughly like:

DRI-kerr (with a short i, like in English drink, and a trilled or tapped r)

Details:

dri-
The dr cluster is similar to English dr in drink, but the r is usually tapped or lightly rolled.
-ck-
The ck is just a hard k sound. Swedish uses ck to mark a preceding short vowel.
-er
Often pronounced a bit like -ehr or -e(r), with a fairly short e and a Swedish r at the end.

There is nothing as tricky as the Swedish sj-sound here; the main challenge is usually the r and keeping the vowel short: dri- with a short i, not like English tree.