Learn Hardcore German: Unsere alte Matratze ist hart, deshalb ist unser Schlaf schlecht. - Our old mattress is hard, therefore our sleep is poor.

Matratze = feminine, singular, nominative
→ you need unsere (feminine nominative form of unser)
→ unsere alte Matratze
Schlaf = masculine, singular, nominative
→ you use the basic form unser (masculine nominative)
→ unser Schlaf

So:

Masculine nominative: unser Vater, unser Schlaf
Feminine nominative: unsere Mutter, unsere Matratze
Neuter nominative: unser Kind
Plural nominative (all genders): unsere Kinder

Why is there no article like die before Matratze or Schlaf? Why not die alte Matratze or der Schlaf?

Possessive words like mein, dein, sein, ihr, unser, euer, Ihr already act like articles.
You normally don’t combine them with another article.

So you say:

unsere Matratze (not die unsere Matratze)
unser Schlaf (not der unser Schlaf)

In other words, unser(e) replaces der/die/das, it does not come on top of it.

Why is it alte Matratze with -e, but Schlaf is just schlecht without -e?

You’re seeing two different types of adjectives:

Attributive adjective (comes before the noun and gets an ending)
- unsere alte Matratze
- alte comes before Matratze, so it must get a case/gender ending (-e here).
Predicative adjective (comes after sein, werden, bleiben, etc., and has no ending)
- ist hart
- ist schlecht

So:

Before the noun → adjective changes:
- eine alte Matratze, unsere alte Matratze
After sein → adjective stays basic:
- Die Matratze ist alt/hart.
- Unser Schlaf ist schlecht.

Why is the word order …, deshalb ist unser Schlaf schlecht and not …, deshalb unser Schlaf ist schlecht?

In German, deshalb is a verb-second connector (a conjunctive adverb). That means:

It counts as being in the first position of the clause.
The finite verb must still be in second position.

So the structure is:

deshalb (position 1)
ist (verb in position 2)
the rest of the sentence (unser Schlaf schlecht)

→ …, deshalb ist unser Schlaf schlecht.

deshalb unser Schlaf ist schlecht is wrong because the verb is no longer in the second position.

What exactly does deshalb mean, and how is it different from weil?

deshalb is a linking word meaning therefore/for that reason/so. It introduces a main clause with normal verb-second word order:

Unsere alte Matratze ist hart, deshalb ist unser Schlaf schlecht.
= Our mattress is hard, therefore our sleep is bad.

weil means because and introduces a subordinate clause with the verb at the end:

Unser Schlaf ist schlecht, weil unsere alte Matratze hart ist.

So:

deshalb → cause in the first clause, result in the second clause, verb in position 2
weil → result in the first clause, cause in the subordinate clause, verb at the end

Why is there a comma before deshalb?

Because German uses commas to separate independent clauses (main clauses).

Your sentence has two main clauses:

Unsere alte Matratze ist hart
deshalb ist unser Schlaf schlecht

Two main clauses joined → you must write a comma before deshalb:

Unsere alte Matratze ist hart, deshalb ist unser Schlaf schlecht.

Why are Matratze and Schlaf in the nominative case here?

They are the subjects of their respective clauses.

In Unsere alte Matratze ist hart,
unsere alte Matratze is the thing that is hard → subject → nominative.
In deshalb ist unser Schlaf schlecht,
unser Schlaf is the thing that is bad → subject → nominative.

In German, the subject of a finite verb is always in the nominative case, so both nouns take nominative forms of unser and of the adjectives.

Could I say Unser Schlaf ist schlecht, weil unsere alte Matratze hart ist instead? Is there any difference?

Yes, that’s perfectly correct German:

Unser Schlaf ist schlecht, weil unsere alte Matratze hart ist.

Main differences:

With deshalb:
- Cause first, result second
- Two main clauses: …, deshalb ist …
- Slightly more “result-focused”: This is true, therefore that happens.
With weil:
- Result first, cause second
- Second clause is a subordinate clause: verb goes to the end (hart ist)
- More “reason-focused”: That happens because this is true.

Meaning is essentially the same; it’s mostly a difference in structure and emphasis.

Why is hart used, not something like harte or harter?

hart comes after ist, so it is a predicative adjective. Predicative adjectives in German do not take endings:

Die Matratze ist hart.
Der Stuhl ist bequem.
Das Essen ist kalt.

You only add endings when the adjective is before a noun:

eine harte Matratze
ein bequemer Stuhl
das kalte Essen

Is schlecht here an adjective or an adverb? In English I would say “our sleep is bad,” but it feels like it could be “we sleep badly.”

Grammatically, schlecht here is an adjective used predicatively:

unser Schlaf (noun)
ist (linking verb)
schlecht (adjective describing the noun)

The structure is exactly like English our sleep is bad.

If you wanted to say we sleep badly, you would normally phrase it differently in German:

Wir schlafen schlecht.
(Here schlecht modifies the verb schlafen, more like an adverb.)

How do I know that Matratze is feminine and Schlaf is masculine? There’s no article der/die/das in the sentence.

You can’t see the gender from this sentence alone; you need to learn the noun with its article:

die Matratze (feminine)
der Schlaf (masculine)

When you know that:

Feminine nominative with unser → unsere Matratze
Masculine nominative with unser → unser Schlaf

General tip: always learn new nouns like this:

die Matratze (f.)
der Schlaf (m.)

Could I say Unsere alte Matratze ist hart, deshalb schlafen wir schlecht? Does that change the meaning?

Yes, that’s a natural and correct variant:

Unsere alte Matratze ist hart, deshalb schlafen wir schlecht.

Differences:

Original: …, deshalb ist unser Schlaf schlecht.
→ Focuses on “our sleep” as a thing that is bad.
Variant: …, deshalb schlafen wir schlecht.
→ Focuses on the action of sleeping and says we sleep badly.

The overall idea (the hard mattress causes poor sleep) is the same; you just shift the emphasis from the noun Schlaf to the verb schlafen.