मैं आज नहीं आ सकती, और मेरी बहन भी नहीं आ सकती।

Because सकती is the feminine form. The speaker is understood to be female.

With सकना (to be able to / can), the form agrees with the subject:

• मैं ... आ सकता हूँ = I can come. (male speaker)
• मैं ... आ सकती हूँ = I can come. (female speaker)

In your sentence, मैं ... आ सकती shows a female speaker, and बहन ... आ सकती is also feminine because बहन is feminine.

Because after a modal verb like सकना, Hindi uses the verb stem, not the infinitive.

So:

• आना = to come
• आ = come (stem used before another verb such as *सकना)*

That is why:

• आ सकती = can come
• जा सकता = can go
• कर सकती = can do

So आना सकती is not correct.

In a neutral Hindi sentence, नहीं usually comes before the verb phrase it negates.

So:

• मैं आज नहीं आ सकती = I cannot come today

This is the most natural, standard placement.

You may sometimes hear other placements for emphasis, but for learners, putting नहीं before the verb phrase is the safest rule.

भी means also, too, or in a negative sentence often either in natural English.

So:

• मेरी बहन भी नहीं आ सकती literally = My sister also cannot come
• natural English = My sister can’t come either

A useful rule: भी usually comes after the word it relates to.

Here:

• मेरी बहन भी = my sister too / my sister as well

They do different jobs:

• और = and
• भी = also / too / as well

So:

• और मेरी बहन भी नहीं आ सकती = and my sister can’t come either

If you removed भी, the sentence would still make sense, but it would lose the idea of also/either:

• और मेरी बहन नहीं आ सकती = and my sister cannot come

That is slightly different from and my sister can’t come either.

Because मेरा / मेरी / मेरे agrees with the thing possessed, not with the owner.

Since बहन is feminine singular, you use मेरी:

• मेरी बहन = my sister

Compare:

• मेरा भाई = my brother
• मेरी बहन = my sister
• मेरे दोस्त = my friends / my friend (depending on context and gender/number)

This agreement does not depend on whether the speaker is male or female.

It can mean either, depending on context. In this sentence, it most naturally means being unable to come in a practical sense — for example, because of time, plans, health, or circumstances.

So मैं आज नहीं आ सकती usually means:

• I can’t come today not necessarily
• I am physically unable to come today

This is very similar to English can’t.

Yes, in conversation it can be omitted if the meaning is clear.

Full version:

• मैं आज नहीं आ सकती, और मेरी बहन भी नहीं आ सकती।

Shorter version:

• मैं आज नहीं आ सकती, और मेरी बहन भी नहीं।

The full version is clearer and is especially good for learners. The shorter version is natural in speech.

Hindi word order is flexible, but the given order is very natural and neutral.

Your sentence:

• मैं आज नहीं आ सकती
  literally follows a pattern like I today not come-can

You could also say:

• आज मैं नहीं आ सकती = Today, I can’t come

Both are correct. The difference is mainly emphasis:

• मैं आज नहीं आ सकती emphasizes I
• आज मैं नहीं आ सकती emphasizes today

For learners, the original order is a very good default.

In fully complete Hindi, you would often say:

• मैं आज नहीं आ सकती हूँ

But in everyday Hindi, the हूँ is very often dropped when it is easy to understand from context:

• मैं आज नहीं आ सकती

Both are correct. The shorter form is extremely common in speech.

The same applies in the second clause:

• मेरी बहन भी नहीं आ सकती (है)

In everyday usage, speakers often leave the auxiliary unstated when it sounds natural.