Hladím štěně.

Hladím is the 1st person singular present tense form of the verb hladit.

So:

• hladit = the dictionary form, to pet / to stroke
• hladím = I pet / I am petting

The ending -ím tells you the subject is I.

Czech often omits subject pronouns when they are clear from the verb ending.

Because hladím already means I am petting, you do not need já.

So:

• Hladím štěně. = normal, natural
• Já hladím štěně. = I am petting the puppy, with extra emphasis or contrast

It can mean both.

In Czech, the present tense of an imperfective verb often covers both:

• a general present: I pet
• an ongoing action: I am petting

The exact meaning depends on context. In many situations, English would naturally translate it as I am petting the puppy.

Štěně is the direct object, so it is in the accusative case.

However, for this noun, the nominative singular and accusative singular look the same:

• nominative: štěně
• accusative: štěně

So even though it is the object, you do not see a visible ending change here.

Štěně is neuter.

That may feel surprising, because it refers to a living creature, but in Czech many words for young beings are neuter, for example:

• štěně = puppy
• kotě = kitten
• dítě = child

This is a normal Czech pattern.

Because štěně belongs to a special group of neuter nouns, especially words for young creatures.

Its base form is štěně, but other cases show a larger stem:

• štěně
• štěněte
• štěňata

So the nominative singular does not always tell you everything about how the noun declines.

Czech has no articles, so there is no direct equivalent of English a/an or the.

So štěně can mean:

• a puppy
• the puppy

Context tells you which is meant.

That is completely normal in Czech.

Yes. Czech word order is fairly flexible.

Hladím štěně. is a natural neutral sentence. But you can also change the order for emphasis:

• Hladím štěně. = neutral
• Štěně hladím. = emphasis on the puppy
• Já hladím štěně. = emphasis on I

So word order in Czech often helps show focus or contrast, not just grammar.

A rough English-friendly approximation is:

HLA-deem SHTYE-nyeh

A few useful details:

• hladím has a long í, so -dím sounds like deem
• ště has no exact English equivalent, but sh-tye is a useful approximation
• ň is a soft n, similar to Spanish ñ
• Czech stress is usually on the first syllable:
  • HLA-dím
  • ŠTĚ-ně

Yes. Hladit is imperfective, so it is used for:

• ongoing action
• repeated action
• general action

That is why it works well in Hladím štěně.

A common perfective partner is pohladit. That usually means something more like:

• to give a stroke / pet once
• a completed act of petting

Compare:

• Hladím štěně. = I am petting the puppy
• Pohladím štěně. = I will pet the puppy / I’ll give the puppy a stroke

So the sentence uses the form that best fits an ongoing present action.