Die Katze will heute nicht gestreichelt werden, aber den Hund kann man immer streicheln.

No. In this sentence, will is from the verb wollen, which means to want.

So:

• Die Katze will ... = The cat wants ...
• not The cat will ... in the future sense

This is a very common false friend for English speakers.

Because the cat is receiving the action, not doing it.

• streicheln = to pet / to stroke
• gestreichelt werden = to be petted / to be stroked

So:

• Die Katze will heute nicht streicheln would mean The cat does not want to pet (someone/something) today
• Die Katze will heute nicht gestreichelt werden means The cat does not want to be petted today

This is a passive infinitive construction:
past participle + werden

So:

• gestreichelt werden = to be petted

Because German forms the passive with:

• past participle
  • werden

Examples:

• streicheln = to pet
• gestreichelt werden = to be petted
• gesehen werden = to be seen
• gelobt werden = to be praised

So in your sentence:

• will ... gestreichelt werden = wants to be petted

If you used streicheln werden, that would not be correct here.

Because heute and nicht are both modifying what follows, and German often places nicht before the element it is negating, or near the end of the part being negated.

Here:

• Die Katze will heute nicht gestreichelt werden

This most naturally means:

• The cat does not want to be petted today

The negation is focused on the whole idea of being petted today.

If you move nicht, the emphasis can change. For example:

• Die Katze will nicht heute gestreichelt werden sounds like not today specifically
• Die Katze will heute nicht gestreichelt werden is the normal, natural version

Because they are in different cases.

• die Katze is the subject of the first clause, so it is in the nominative
• den Hund is the direct object of the second clause, so it is in the accusative

Here is the breakdown:

• Die Katze = nominative feminine singular
• den Hund = accusative masculine singular

English does not mark this with articles, but German does.

Yes, man kann den Hund immer streicheln is also correct.

German allows you to move elements to the first position for emphasis or topic. This is often called fronting or topicalization.

So:

• Den Hund kann man immer streicheln
• Man kann den Hund immer streicheln

Both are grammatical.

The version in your sentence puts a little more focus on the dog, especially as a contrast with the cat:

• The cat doesn’t want to be petted today, but the dog can always be petted.

Because German main clauses follow the verb-second rule.

That means the finite verb must be in the second position.

In the clause:

• Den Hund kann man immer streicheln

the first position is occupied by Den Hund, so the finite verb kann must come next.

Structure:

• Position 1: Den Hund
• Position 2: kann
• then: man immer streicheln

So den Hund man kann is not correct in a normal main clause.

man is an indefinite pronoun meaning something like:

• one
• people
• you in a general sense

So:

• Den Hund kann man immer streicheln

means:

• One can always pet the dog
• You can always pet the dog
• People can always pet the dog

It does not mean man as in male person.

Because man is grammatically singular in German.

So it takes singular verb forms:

• man kann
• man will
• man sagt

not:

• man können
• man wollen
• man sagen

Even though it can refer to people in general, the grammar is singular.

Because kann is a modal verb, and with modal verbs in German, the main verb usually goes to the end of the clause in the infinitive.

So:

• kann ... streicheln
• will ... werden

Examples:

• Ich kann Deutsch sprechen.
• Sie will nach Hause gehen.
• Man kann den Hund immer streicheln.

That is why streicheln appears at the end.

Because the first clause contains both:

• a modal verb: will
• a passive infinitive: gestreichelt werden

With a modal verb, the infinitive part goes to the end.
The passive infinitive itself is:

• gestreichelt werden

So the whole structure is:

• Die Katze will heute nicht gestreichelt werden

Literally, the end is built like:

• want ... petted be

This sounds strange in English, but it is normal in German.

It can mean both, depending on context.

Common meanings:

• to pet an animal
• to stroke someone or something gently

In this sentence, because we are talking about a cat and a dog, pet is the most natural English translation.

So:

• die Katze streicheln = to pet the cat
• den Hund streicheln = to pet the dog

Aber means but and introduces a contrast between the two clauses.

The contrast is:

• The cat does not want to be petted today
• but the dog can always be petted

It connects two main clauses, and the verb in the second clause still follows normal main-clause word order:

• ..., aber den Hund kann man immer streicheln.

Yes. That is also correct.

• will from wollen is more direct: wants
• möchte is often softer or more polite: would like

So:

• Die Katze will heute nicht gestreichelt werden = stronger, more direct
• Die Katze möchte heute nicht gestreichelt werden = softer in tone

With an animal, will sounds perfectly natural.

Because the sentence has two coordinated main clauses:

1. Die Katze will heute nicht gestreichelt werden
2. aber den Hund kann man immer streicheln

In German, a comma is normally used to separate such clauses when they are joined by conjunctions like aber.

So the comma helps show the structure clearly.

Yes, absolutely. German often uses this kind of wording naturally, especially with pets.

• Die Katze will heute nicht gestreichelt werden sounds like: the cat is not in the mood
• den Hund kann man immer streicheln suggests the dog is always happy to be petted

So even though animals are being described with human-like intention, the sentence sounds normal and idiomatic.