Biorę smycz i idę z psem na spacer.

Because Polish often drops subject pronouns when they are already clear from the verb ending. Both biorę and idę are 1st person singular, so the sentence already means I take the leash and I go for a walk with the dog.

You would add ja only for emphasis or contrast, for example: Ja biorę smycz, a ty otwierasz drzwi.
= I take the leash, and you open the door.

Biorę is the present tense of brać, an imperfective verb. Here it means I take, I’m taking, or I pick up.

It fits because the speaker is describing an action happening now or as part of a current sequence.

Wezmę is from the perfective verb wziąć and usually means I will take or I’ll take in the sense of one completed action in the future.

So:

• Biorę smycz = I’m taking the leash.
• Wezmę smycz = I’ll take the leash.

The verb brać takes a direct object in the accusative case, so smycz is indeed in the accusative here.

The reason it does not visibly change is that this type of feminine noun has the same form in nominative and accusative singular.

So:

• nominative: smycz
• accusative: smycz

Even though the form looks the same, the case is different because of its role in the sentence.

This is a very common question.

• idę (from iść) is used for one specific trip, usually happening now or in the immediate situation.
• chodzę (from chodzić) is used for habitual, repeated, or more general movement on foot.

So:

• Idę z psem na spacer = I’m going for a walk with the dog now / on this occasion.
• Chodzę z psem na spacery = I go on walks with the dog regularly.

In this sentence, the speaker is talking about a specific walk, so idę is the natural choice.

Because the preposition z meaning with requires the instrumental case.

The noun pies changes like this:

• nominative: pies = dog
• genitive/accusative: psa
• instrumental: psem

So:

• z psem = with the dog

That is why pies becomes psem here.

No. Z can mean different things depending on context, and the case after it helps you understand which meaning it has.

Two very common uses are:

• z + instrumental = with
  z psem = with the dog
• z + genitive = from / out of / off
  z domu = from the house

So in z psem, the instrumental form psem shows that the meaning is with the dog, not from the dog.

Because na spacer is the standard expression meaning for a walk.

Here na is used with the accusative to show movement toward a goal or activity:

• iść na spacer = to go for a walk

By contrast, na spacerze is locative and means being on a walk or during a walk:

• Jestem na spacerze. = I’m out on a walk.

So:

• idę na spacer = I’m going for a walk
• jestem na spacerze = I’m on a walk

Because Polish does not use articles like English a and the.

So smycz can mean:

• a leash
• the leash

And pies/psem can mean:

• a dog
• the dog

The exact meaning usually comes from context. If Polish needs to be more specific, it uses other words, for example:

• ta smycz = this/the leash
• mój pies = my dog

Yes. Polish word order is more flexible than English word order because grammatical endings show the role of many words.

The sentence
Biorę smycz i idę z psem na spacer.
is a neutral, natural order.

Other orders are possible, but they change the emphasis. For example:

• Z psem idę na spacer. = emphasis on with the dog
• Na spacer idę z psem. = emphasis on for a walk

So the original version is a good basic, neutral way to say it.

Most naturally, it describes a current or immediate action: the speaker is taking the leash and going out now.

That feeling comes especially from idę, which usually refers to a specific trip.

Polish present tense can also describe habits, but for a general repeated action you would more often see something like:

• Chodzę z psem na spacery. = I go on walks with the dog.
• Codziennie biorę smycz i idę z psem na spacer. = Every day I take the leash and go for a walk with the dog.

So without extra context, this sentence sounds like what I’m doing now.