Denne skjorte er lavet af bomuld, men min gamle trøje er lavet af uld.

Denne means this.

In standard Danish, denne skjorte = this shirt.
But in everyday spoken Danish, many people more often say den her skjorte.

A useful pattern is:

• denne / den her for a common-gender singular noun
• dette / det her for a neuter singular noun
• disse / de her for plurals

So denne skjorte is correct, but den her skjorte is often more conversational.

Because after words like min, din, den, denne, and other determiners, adjectives usually take the -e form in Danish.

So:

• en gammel trøje = an old sweater
• min gamle trøje = my old sweater

This is often called the weak adjective form. It appears when the noun is made definite in some way, including by a possessive such as min.

Because trøje is a common-gender noun.

In Danish:

• min is used with common-gender singular nouns
• mit is used with neuter singular nouns
• mine is used with plurals

So:

• min trøje = my sweater
• mit hus = my house
• mine sko = my shoes

Both skjorte and trøje are common-gender nouns, so forms like denne skjorte and min trøje are expected.

Er lavet af means is made of or is made from.

• er = is
• lavet = made
• af = of / from

So:

• Denne skjorte er lavet af bomuld = This shirt is made of cotton
• Min gamle trøje er lavet af uld = My old sweater is made of wool

In everyday Danish, lavet af is a very common way to talk about what something is made from.

For material or composition, Danish normally uses af.

So you say:

• lavet af bomuld
• lavet af uld
• lavet af træ

Fra usually means from in the sense of origin, source, or starting point:

• Jeg er fra England = I am from England
• Brevet er fra min ven = The letter is from my friend

A helpful rule is: when you mean made of/from a material, use af.

Because bomuld and uld are being used as materials in a general, uncountable sense.

That is similar to English:

• made of cotton
• made of wool

Not:

• made of a cotton
• made of a wool

So in Danish, bare material nouns are normal here:

• af bomuld
• af uld

If you were talking about a specific wool or a specific cotton, the structure could be different, but in this sentence the general material is meant.

Skjorte usually means shirt, especially the kind with a collar and often buttons.

Trøje is broader. It often means something like:

• sweater
• jumper
• pullover
• sometimes just a general top

So in this sentence, skjorte and trøje are two different kinds of clothing, which is why the contrast with men makes sense.

No. Men does not cause inversion here.

Men is a coordinating conjunction, like but in English. It joins two main clauses, and each clause keeps normal main-clause word order.

So:

• Denne skjorte er lavet af bomuld
• men min gamle trøje er lavet af uld

In the second clause, the order is still:

subject + verb + rest

So:

• min gamle trøje = subject
• er = verb

In normal careful Danish, repeating it is completely natural and often the clearest choice:

• Denne skjorte er lavet af bomuld, men min gamle trøje er lavet af uld.

Sometimes in speech or informal writing, people may omit repeated words if the meaning is obvious, but for learners it is best to keep the full structure.

So yes, repeating er lavet af is the safest and most natural way to say it.

There are other options, but lavet af is the most everyday one.

Some alternatives are:

• fremstillet af = manufactured/produced from, more formal
• består af = consists of, more factual or technical

Examples:

• Skjorten er fremstillet af bomuld
• Trøjen består af uld

But for normal conversation, er lavet af is usually the most natural choice.

Yes, gammel usually means old, and here gamle describes the sweater as being old.

So min gamle trøje means my old sweater.

It normally refers to age or how long you have had it, but like English old, it can sometimes also suggest familiarity or long use depending on context.

In this sentence, the simplest reading is just that the sweater is an older one.