Dnes nemám moc času.

A word-by-word breakdown is:

• Dnes = today
• nemám = I do not have
• moc = much / a lot
• času = of time

So the structure is literally something like Today I do not have much of time, but the natural English meaning is I don’t have much time today.

Nemám comes from the verb mít = to have.

It is the 1st person singular form, so:

• mám = I have
• nemám = I do not have

The ne- prefix makes the verb negative, which is very common in Czech:

• mám čas = I have time
• nemám čas = I don’t have time

Because Czech usually does not need subject pronouns unless you want emphasis or contrast.

The ending of the verb already tells you who the subject is:

• mám = I have
• máš = you have
• má = he/she/it has

So nemám already means I don’t have, and já would only be added for emphasis, for example:

• Já dnes nemám moc času. = I don’t have much time today
  (with extra stress on I)

Because after words of quantity like moc, Czech normally uses the genitive form of the noun.

So:

• moc času = much time
• hodně práce = a lot of work
• málo peněz = little money

The basic dictionary form is čas, but after moc it becomes času.

This is one of the most important patterns to learn:

• quantity word + noun in the genitive

It can mean different things depending on what it modifies.

In this sentence, moc means much / a lot:

• moc času = much time

But with adjectives or adverbs, moc often means very:

• moc dobrý = very good
• moc rychle = very quickly

So here it is a quantity word, not an intensifier.

Yes. That is also natural Czech.

Czech word order is more flexible than English, and the choice often depends on emphasis or information flow.

Compare:

• Dnes nemám moc času.
  Focuses first on today
• Nemám dnes moc času.
  Starts with I don’t have
• Moc času dnes nemám.
  Puts extra emphasis on much time

All of these can work, but Dnes nemám moc času is a very normal, natural sentence.

Yes, mostly.

• dnes = today
• dneska = also today

Dneska sounds a bit more conversational and informal in many contexts.
Dnes is short, neutral, and very common.

So:

• Dnes nemám moc času.
• Dneska nemám moc času.

Both are natural.

Yes, but the nuance is a little different.

• nemám moc času = very common, natural, conversational
• nemám mnoho času = correct, but more formal or bookish
• nemám hodně času = understandable, but with negation Czech often prefers moc here

For everyday speech, nemám moc času is probably the most natural choice.

It is neutral everyday Czech.

It is not slang, and it is not especially formal either. You can use it in many situations:

• with friends
• at work
• in polite conversation
• in writing a message

If you wanted a more formal version, you might choose different phrasing depending on context, but this sentence itself is completely standard and natural.

A simple learner-friendly approximation is:

• Dnes ≈ dnyes
• nemám ≈ neh-maam
• moc ≈ mots
• času ≈ cha-su

A few useful notes:

• dnes begins with a consonant cluster, so it may feel a little tight for English speakers.
• á in nemám is a long vowel.
• č in času sounds like ch in church.
• c in moc sounds like ts.

The stress in Czech is usually on the first syllable of each word:

• DNES
• NEmám
• MOC
• ČAsu

Yes.

• Nemám čas. = I don’t have time.
• Nemám moc času. = I don’t have much time.

The second one is a bit more specific: it says you do have some time, but not a lot.

So:

• Nemám čas can sound stronger, like I have no time / I’m not available
• Nemám moc času often means I’m short on time

Both are very useful, but they are not exactly the same.