हम आज नहीं दौड़ रहे हैं।

• हम = we
• आज = today
• नहीं = not
• दौड़ = from दौड़ना, to run
• रहे हैं = the progressive part, roughly are ...-ing

So the whole sentence means We are not running today.

In Hindi, नहीं usually comes before the part being negated.

So in:

• हम आज नहीं दौड़ रहे हैं

the not applies to the action running.

This is very normal Hindi word order:

• मैं नहीं जा रहा हूँ = I am not going
• वह नहीं पढ़ रही है = She is not studying

English puts not after the helping verb (are not running), but Hindi usually puts नहीं before the main verb phrase.

This is the present progressive (also called the present continuous).

It shows an action that is:

• happening now, or
• being described as ongoing around the present time

Structure:

• verb stem + रहा / रही / रहे + होना

Here:

• दौड़ = verb stem
• रहे = progressive participle
• हैं = present form of to be

So:

• दौड़ रहे हैं = are running

With नहीं, it becomes:

• नहीं दौड़ रहे हैं = are not running

Because हम is grammatically plural, and with a plural or masculine/mixed group subject, Hindi uses रहे हैं in the progressive.

Compare:

• मैं दौड़ रहा हूँ = I am running (masculine speaker)
• मैं दौड़ रही हूँ = I am running (feminine speaker)
• हम दौड़ रहे हैं = We are running

So रहे agrees with हम.

Usually, no. If we refers to an all-female group, you would normally say:

• हम आज नहीं दौड़ रही हैं।

Why?

• रही is the feminine form
• हैं stays the same because the subject is still we

So:

• mixed group / masculine group: हम ... दौड़ रहे हैं
• all-female group: हम ... दौड़ रही हैं

हैं is the present-tense form of होना (to be) used with हम / आप / वे.

In the progressive, Hindi normally uses:

• verb stem + रहा / रही / रहे
  • form of होना

So:

• दौड़ रहे हैं literally works like running are

Without हैं, the sentence would sound incomplete in standard Hindi.

In casual speech, people may sometimes shorten or blur the ending, but in correct standard Hindi, हैं belongs there.

Yes, but only in certain special contexts.

Normally:

• हम = we

But sometimes हम can mean I:

• in some regional speech
• in older/literary styles
• as a humble or formal self-reference in some contexts

For most learners, though, in a sentence like this, the safest reading is simply:

• हम = we

The dictionary form is:

• दौड़ना = to run

In many Hindi verb constructions, you remove -ना and use the stem:

• दौड़ना → दौड़

Then you add the needed pattern:

• दौड़ रहा है
• दौड़ रही है
• दौड़ रहे हैं

So दौड़ here is just the verb stem used in the progressive construction.

Hindi word order is somewhat flexible, but not every variation sounds equally natural.

The most natural version here is:

• हम आज नहीं दौड़ रहे हैं।

This places:

1. subject
2. time word
3. negation
4. verb phrase

You might hear other orders for emphasis, but हम नहीं आज दौड़ रहे हैं sounds awkward in normal conversation.

A few natural possibilities are:

• हम आज नहीं दौड़ रहे हैं।
• आज हम नहीं दौड़ रहे हैं।

The second one emphasizes today a bit more.

Good question. These two sentences are related, but not identical.

• हम आज नहीं दौड़ रहे हैं = We are not running today

  • progressive / ongoing situation
  • often about what is happening or what we are doing today

• हम आज नहीं दौड़ते हैं = literally We do not run today

  • this sounds less natural in many contexts
  • दौड़ते हैं is habitual, so it usually describes a habit or repeated action

More natural habitual examples:

• हम रोज़ दौड़ते हैं। = We run every day.
• हम रविवार को नहीं दौड़ते हैं। = We do not run on Sundays.

So for today as a current situation, नहीं दौड़ रहे हैं is usually the better choice.

It can often suggest either, depending on context.

Most directly, it means:

• We are not running today

In real use, that can imply:

• we are not running at the moment, or
• we are not doing our run today / not going running today

Because आज sets the time frame as today, the sentence often feels broader than just one exact moment.

So the exact nuance depends on the situation, but the grammar itself is completely normal for We are not running today.

Yes. दौड़ना (to run) is normally intransitive, which means it does not need a direct object.

So:

• हम दौड़ रहे हैं = complete sentence
• no object is required

Just like in English, run can stand alone:

• We are running

That is why the Hindi sentence does not need anything after the verb.