Chcemy się targować i negocjować niższą cenę.

The idea of we is built into chcemy.

• chcieć = to want
• chcemy = we want

Polish verb endings usually show the subject, so you do not need a separate word for we. You could add my for emphasis, but normally it is omitted:

• (My) chcemy się targować... = We want to haggle...

So chcemy already tells you it is 1st person plural: we.

After chcieć (to want), Polish normally uses the infinitive directly:

• Chcę jeść. = I want to eat.
• Chcemy rozmawiać. = We want to talk.

So in your sentence:

• chcemy = we want
• się targować = to haggle
• negocjować = to negotiate

English uses to before the verb, but Polish does not add a separate word there. The infinitive form itself does the job.

Here, się is part of the verb targować się.

For a learner, the safest way to remember it is:

• targować się = to haggle / bargain

In this sentence, się does not literally mean oneself. Very often in Polish, się is just part of how a verb works.

So you should learn these as two different verb patterns:

• targować się = haggle, bargain
• negocjować = negotiate

You cannot normally just remove się and keep the same meaning.

Yes, that is possible, but Chcemy się targować... is the more neutral and more natural order for many speakers.

Polish word order is fairly flexible, and short words like się often appear early in the clause:

• Chcemy się targować...
  • very natural
• Chcemy targować się...
  • also possible

So both can be grammatical, but if you are unsure, putting się before the infinitive is a good default in a sentence like this.

They are similar, but not identical.

• targować się = to haggle / bargain

  • often about price
  • often more informal
  • common in markets, shops, private sales

• negocjować = to negotiate

  • broader meaning
  • can be more formal
  • can apply to price, contract terms, conditions, deadlines, and so on

So in everyday English, targować się is closer to haggle, while negocjować is closer to negotiate.

Because the phrase is in the accusative case, not the nominative.

The basic dictionary forms are:

• niższa cena = a lower price (subject form / nominative)

But after a verb like negocjować, you need the direct object, so Polish uses the accusative:

• niższą cenę

The endings change like this:

• niższa → niższą
• cena → cenę

So:

• To jest niższa cena. = This is a lower price.
• Negocjujemy niższą cenę. = We are negotiating a lower price.

Yes. Niższą cenę is the more standard and natural phrase.

In Polish, people normally say:

• niższa cena = a lower price

The adjective tańszy / tańsza / tańsze is more naturally used with the thing that costs less:

• tańszy produkt = a cheaper product
• tańszy hotel = a cheaper hotel

So:

• negocjować niższą cenę sounds natural
• negocjować tańszą cenę sounds less idiomatic

This is a very good question. The sentence is understandable, but many speakers would expect targować się o + something.

A very natural pattern is:

• targować się o niższą cenę = to haggle over / for a lower price

So these are especially natural:

• Chcemy się targować o niższą cenę.
• Chcemy negocjować niższą cenę.

Your original sentence is understandable, but it slightly compresses two different verb patterns into one shared object. That is why some learners may feel it is a bit awkward.

This is about aspect.

• negocjować = imperfective

  • focuses on the process
  • to negotiate / be negotiating

• wynegocjować = perfective

  • focuses on the result
  • to negotiate successfully / to secure by negotiation

So:

• Chcemy negocjować niższą cenę.
  = We want to negotiate a lower price.
  Focus: the activity/process.

• Chcemy wynegocjować niższą cenę.
  = We want to get/negotiate a lower price successfully.
  Focus: the outcome.

Both can be correct, but they do not mean exactly the same thing.

Usually, no.

Polish normally leaves subject pronouns out unless they are needed for emphasis, contrast, or clarity. Since chcemy already means we want, my is optional.

• Chcemy się targować... = normal, neutral
• My chcemy się targować... = more emphatic, like we want to haggle

So in ordinary speech, the version without my is completely normal.