Learn Hardcore German: Am Sonntag wollen meine Mitbewohnerin und ich etwas Neues ausprobieren und haben eine Verabredung zum Tanzen. - On Sunday my roommate and I want to try something new and have a date to go dancing.

First element here: Am Sonntag (“On Sunday”) – this whole time phrase counts as one element.
Second element must be the finite verb: wollen.
The subject meine Mitbewohnerin und ich comes after that.

Basic order without the time phrase would be:

Meine Mitbewohnerin und ich wollen etwas Neues ausprobieren.

When you move the time phrase to the front, you get:

Am Sonntag wollen meine Mitbewohnerin und ich etwas Neues ausprobieren.

This “inversion” (verb before subject) is normal and very common in German whenever something other than the subject is put first.

Why is it Am Sonntag and not an Sonntag, im Sonntag, or something else?

Am is a contraction of an + dem (preposition an + dative article dem).

For days of the week and dates, German normally uses an in the dative:

an dem Sonntag → am Sonntag = “on Sunday”
am Montag, am Dienstag, am 1. Mai, etc.

Some contrasts:

am Sonntag – on Sunday (a specific day)
im Sommer – in (the) summer (seasons, months: im Januar, im Herbst)
in 2025 – in 2025 (years usually with in)

So am Sonntag is simply the standard way to say “on Sunday”.

Why is it meine Mitbewohnerin und ich and how does the verb agree with that?

Meine Mitbewohnerin und ich is the subject of the sentence. It’s made up of:

meine Mitbewohnerin – “my (female) flatmate / roommate”
ich – “I”

Together they mean “my flatmate and I”, which is grammatically the same as “wir” (“we”).
Because the subject is effectively we, the verb takes the 1st person plural form:

wir wollen → meine Mitbewohnerin und ich wollen
wir haben → meine Mitbewohnerin und ich haben

Word order:

Meine Mitbewohnerin und ich wollen … – neutral order.
Ich und meine Mitbewohnerin wollen … – grammatically OK, but sounds a bit more childish or less polite; German (like English) usually puts the other person before “I”.

In your sentence, the subject is just placed after the verb because of the verb-second rule, but it’s the same subject:

Am Sonntag wollen meine Mitbewohnerin und ich …

Why is ausprobieren at the very end of the first part of the sentence?

You have a modal verb plus another verb:

wollen … ausprobieren = “want to try out”

In German, with a modal verb:

The modal (wollen) is conjugated and goes in second position.
The other verb (ausprobieren) stays in the infinitive and normally goes to the end of the clause.

That gives:

Am Sonntag wollen … etwas Neues ausprobieren.

Also, ausprobieren is a separable verb: aus- (prefix) + probieren (to try).
Without a modal, in a simple present-tense sentence, it would split:

Am Sonntag probieren wir etwas Neues aus.

With a modal (wollen, können, müssen, etc.), the verb stays unsplit at the end:

Am Sonntag wollen wir etwas Neues ausprobieren.

What is the difference between ausprobieren, probieren, and versuchen here?

All three can be translated with “to try”, but they’re used differently:

ausprobieren
“to try out / test / experiment with (something new)”
Focus: seeing what something is like by using or doing it.
→ etwas Neues ausprobieren = “try out something new” (a new activity, hobby, restaurant, etc.)
probieren
Very often used for tasting food or drink, or briefly trying something.
→ Möchtest du den Kuchen probieren? – “Do you want to taste the cake?”
versuchen
“to attempt / make an effort to do something”
Focus: the attempt, not the novelty.
→ Ich versuche, Deutsch zu lernen. – “I’m trying to learn German.”

In your sentence, ausprobieren is best, because it’s about trying out a new activity (dancing).

Why is Neues capitalised in etwas Neues ausprobieren?

Here Neues is actually a noun, not just an adjective.

Structure:

etwas = “something”
Neues = “new (thing)” → an adjective turned into a noun

German often takes an adjective and “nominalises” it (turns it into a noun). When that happens:

It is capitalised.
It gets a case/gender ending like an adjective.

So:

etwas Neues = “something new” (literally “something new-NEUT.SG”)
nichts Gutes = nothing good
alles Gute = all the best
etwas Schönes = something nice

That’s why you see Neues with a capital N and the ending -es.

What exactly does Verabredung mean, and is it always romantic?

Verabredung means “an arrangement to meet” / “a (social) appointment”. It’s usually:

informal / private (with friends, partners, family)
a plan that two or more people have made with each other

It can mean a romantic date, especially in certain contexts, but doesn’t have to.
Compare:

eine Verabredung mit Freunden – a plan/meet-up with friends
eine Verabredung mit meinem Freund / meiner Freundin – often understood as a date

Other related words:

der Termin – a formal or fixed appointment (doctor, office, official meeting)
das Treffen – a (more neutral) meeting/get-together

So eine Verabredung zum Tanzen is “a planned meet-up / (possibly) date to go dancing”, and whether it’s romantic depends on context.

Why is it zum Tanzen and not zu tanzen or für das Tanzen?

zum Tanzen is:

zu + dem → zum (preposition + dative article)
Tanzen – a nominalised infinitive (verb turned into a noun: “dancing”)

This pattern zum + Verb-noun is very common for expressing purpose:

Zeit zum Lesen – time for reading
Lust zum Tanzen – desire to dance
eine Verabredung zum Tanzen – a (planned) meeting for dancing

Why not zu tanzen?
That would be an infinitive construction, and you’d usually need a different structure:

Wir haben eine Verabredung, tanzen zu gehen. – We have a date to go dancing.

Why not für das Tanzen?
für das Tanzen is grammatically possible but sounds less natural here and feels more like “for the benefit of dancing (as an activity in general)” rather than “to go dance together”. For a specific planned activity, zum Tanzen is the idiomatic choice.

Why is haben in the middle (… und haben eine Verabredung …) and not at the end of the sentence?

You have two coordinated predicates (two things the same subject will do/have):

wollen … ausprobieren – want to try something new
haben eine Verabredung … – have a date/appointment …

So the structure is:

Am Sonntag (time)
wollen (finite verb 1, in second position)
meine Mitbewohnerin und ich (subject)
etwas Neues ausprobieren (rest of predicate 1)
und haben (conjunction + finite verb 2)
eine Verabredung zum Tanzen (rest of predicate 2)

Only one finite verb has to obey the strict “second position” rule – that’s wollen.
The second finite verb (haben) belongs to the second part of the coordination and comes before its object, just like in a simple sentence:

Wir haben eine Verabredung.

You could also say:

… und eine Verabredung zum Tanzen haben.

That’s also correct and perhaps even slightly more common. German allows both orders here; the key point is that haben is a normal main-clause finite verb and does not need to go to the final position (there’s no extra infinitive or participle after it).

Why is the present tense used (wollen, haben) if this is about the future?

German very often uses the present tense + a time expression to talk about the future. Your sentence has:

Am Sonntag – clearly marks the time as future (this coming Sunday)
wollen, haben – present tense forms

This is completely normal and usually preferred:

Am Sonntag wollen wir etwas Neues ausprobieren.
= “On Sunday we’re going to try something new.”

German futur I with werden exists:

Am Sonntag werden wir etwas Neues ausprobieren.

…but it’s used more for:

making predictions,
emphasising the future,
or when there’s no clear time reference.

In everyday speech, for planned future actions with a time phrase like am Sonntag, the present tense is the default.