Ця тканина нова, тоді як підкладка стара.

Because ця has to agree with тканина in gender, number, and case.

• тканина is a feminine singular noun
• the nominative singular feminine form of this is ця

So:

• цей = masculine
• ця = feminine
• це = neuter

That is why ця тканина means this fabric.

They are adjectives agreeing with feminine singular nouns.

• тканина is feminine singular, so нова must also be feminine singular
• підкладка is feminine singular, so стара must also be feminine singular

Compare:

• новий = new, masculine
• нова = new, feminine
• нове = new, neuter

And similarly:

• старий = old, masculine
• стара = old, feminine
• старе = old, neuter

So the endings show grammatical agreement.

In Ukrainian, the present-tense verb to be is usually omitted.

So:

• Ця тканина нова literally looks like This fabric new
• but it naturally means This fabric is new

The same happens in the second part:

• підкладка стара = the lining is old

This is completely normal Ukrainian grammar. If you included a present-tense is, it would usually sound unnatural in an ordinary sentence like this.

тоді як is a conjunction meaning whereas, while, or whereas on the other hand.

In this sentence, it marks a contrast:

• Ця тканина нова = this fabric is new
• тоді як підкладка стара = whereas the lining is old

It is a slightly more explicit contrast than just using and. It helps show that the two parts are being compared.

Yes, in many situations you could say:

• Ця тканина нова, але підкладка стара.

That means This fabric is new, but the lining is old.

The difference is mainly in nuance:

• але = but
• тоді як = whereas / while, often sounding a bit more comparative or formal

So тоді як emphasizes contrast between two parallel facts.

Підкладка means lining, especially the inner layer of a piece of clothing, a bag, or something similar.

So in this sentence:

• тканина = the outer fabric/material
• підкладка = the inner lining

This makes the sentence sound like it is talking about a garment or another sewn item.

Because тоді як introduces another clause, and Ukrainian normally uses a comma before conjunctions that connect clauses like this.

The sentence has two parts:

• Ця тканина нова
• підкладка стара

They are linked by тоді як, so a comma is needed:

• Ця тканина нова, тоді як підкладка стара.

This is standard punctuation.

Yes. Both are in the nominative singular because each one is the subject of its clause.

• Ця тканина нова → тканина is the subject
• підкладка стара → підкладка is the subject

The adjectives and demonstrative also match that nominative form:

• ця
• нова
• стара

Yes, but it changes the structure.

• нова тканина = new fabric
  This is a noun phrase, where the adjective directly describes the noun.
• тканина нова = the fabric is new
  This is the predicate structure used in the sentence.

So:

• Ця нова тканина... = This new fabric...
• Ця тканина нова... = This fabric is new...

Both are correct, but they do different jobs.

Not completely. Ukrainian word order is more flexible than English, but the original version is the most neutral and natural here:

• Ця тканина нова, тоді як підкладка стара.

You may see variations for emphasis, for example:

• Підкладка стара, тоді як ця тканина нова.

That would shift attention toward the lining first.

Even though word order can change, the adjective endings still show the relationships clearly.

Usually the contrast gets the emphasis:

• Ця тканина нова, тоді як підкладка стара.

A speaker will often stress нова and стара, because those are the contrasting ideas: new versus old.

Depending on context, they might also stress:

• ця if contrasting this fabric with another one
• підкладка if contrasting the lining with the fabric

So the stress depends on what the speaker wants to highlight, but the most natural contrast is between нова and стара.