Стоит сестре сесть за стол, как она берёт кисточку и начинает рисовать новый рисунок.

This is an idiomatic pattern meaning as soon as sister sits down at the table, ... or no sooner does she sit down than....

The structure is:

стоит + dative noun/pronoun + perfective infinitive, как + main clause

So here:

• стоит
• сестре
• сесть
• как она берёт...

Together, it expresses that one action happens immediately after another.

Neither in the literal sense.

Here стоит is part of a fixed expression: стоит кому-то что-то сделать, как.... In this pattern, it means something like it only takes someone to do X for Y to happen.

So it is best learned as a whole construction, not word-for-word.

Because this construction requires the person who does the infinitive action to be in the dative:

• стоит мне войти, как... — as soon as I come in...
• стоит ему сказать, как... — as soon as he says...

So сестре is the dative form of сестра, and it marks the person who performs сесть.

Because сесть is perfective, and here Russian wants the idea of a single completed action: to sit down.

• сесть = to sit down, to take a seat
• садиться = to be sitting down / to sit down repeatedly / in process

In the pattern стоит кому-то + perfective infinitive, the perfective is very common because it marks the triggering event: once she has sat down, the next action happens.

Because сесть за стол is the normal Russian expression for to sit down at the table.

There is an important contrast:

• сесть за стол — to sit down at the table
• сидеть за столом — to be sitting at the table

With verbs of motion like сесть, Russian often uses за + accusative to show movement into position.
That is why we get за стол here.

They are in the present tense because the sentence describes a habitual or typical sequence of actions.

It is like saying in English:

• Whenever she sits down at the table, she takes a brush and starts drawing...

Russian often uses the present tense for repeated, general, or vivid narrative actions.

Because after начинать (to begin/start), Russian normally uses an imperfective infinitive.

• рисовать = to draw, to be drawing
• нарисовать = to draw and finish, to complete

When you start doing something, Russian focuses on the process that is beginning, not on its completion. So начинает рисовать is the natural choice.

Here кисточка means a brush, most likely a paintbrush or an artist’s brush.

The ending -очк- is a diminutive suffix. Diminutives in Russian do not always sound especially cute in English; very often they are just normal everyday words. So кисточка can simply mean brush in a natural way.

Because it is the direct object of берёт (takes), so it is in the accusative case.

The dictionary form is кисточка.
The accusative singular is кисточку.

So:

• кисточка — nominative
• кисточку — accusative

Yes, it is understandable and grammatically fine.

Russian can sometimes use a verb and noun from the same root, especially when the noun means the result of the action:

• рисовать рисунок — draw a drawing/picture

That said, stylistically it can sound a bit repetitive. Depending on context, a speaker might also say:

• рисовать картину
• рисовать картинку
• simply рисовать

But рисовать новый рисунок is not wrong.

Yes, it could often be omitted.

Russian frequently drops subject pronouns when the verb ending already makes the subject clear. So a version like:

Стоит сестре сесть за стол, как берёт кисточку и начинает рисовать новый рисунок.

can be understood from context.

However, она is perfectly natural here too. It adds clarity and slightly strengthens the link back to сестре.

Russian has no articles, so there is no separate word for a or the.

A bare noun like сестра / сестре can mean a sister, the sister, or my sister, depending on context. If the speaker wants to be explicit, Russian can add a possessive:

• моей сестре — to my sister
• его сестре — to his sister

Without that, the exact meaning comes from the situation or from the translation already given.