Я принесла стремянку, чтобы повесить новую полку в коридоре.

Принесла is past tense, singular, feminine. It tells you that the speaker is female.

• я принесла = I brought (said by a woman)
• я принёс / принес = I brought (said by a man)

In Russian past tense, verbs agree with gender in the singular:

• masculine: принёс / принес
• feminine: принесла
• neuter: принесло

So this sentence specifically sounds like a woman saying it.

Yes, Russian often drops subject pronouns when they are clear from the verb form.

So both are possible:

• Я принесла стремянку...
• Принесла стремянку...

Because принесла already shows 1st person singular and feminine, the listener can usually understand who did it.
Including я can make the subject clearer, more explicit, or slightly more emphatic.

Стремянка means a stepladder or folding ladder.

In this sentence it becomes стремянку because it is the direct object of принесла: it is the thing that was brought.

• dictionary form: стремянка
• accusative singular: стремянку

This is normal for many feminine nouns ending in -а:

• книга → книгу
• лампа → лампу
• стремянка → стремянку

Чтобы means in order to / so that and introduces a purpose.

So:

• Я принесла стремянку, чтобы повесить новую полку... = I brought a stepladder in order to hang a new shelf...

Russian often uses чтобы + infinitive when the subject is the same in both parts of the sentence.

Here, the same person:

• brought the stepladder
• wants to hang the shelf

So the infinitive повесить works naturally.

If the subject were different, Russian would usually use a finite verb:

• Я принесла стремянку, чтобы он повесил полку.
• I brought a stepladder so that he could hang the shelf.

Повесить is a perfective verb. It focuses on the action as a completed whole or on the result.

Here that makes sense because the purpose is to get the shelf hung/installed, not just to be in the process of hanging it.

Compare:

• вешать = imperfective, process/repeated action
• повесить = perfective, complete the action once

In this sentence, чтобы повесить новую полку suggests a concrete goal: to put up the new shelf successfully.

Both words are in the accusative singular feminine because полка is the direct object of повесить.

• dictionary form: новая полка
• accusative: новую полку

This is a very common pattern:

For feminine singular adjectives ending in -ая, the accusative usually becomes -ую:

• новая → новую

For feminine singular nouns ending in -а, the accusative usually becomes -у:

• полка → полку

So:

• повесить новую полку = to hang a new shelf

Because here it means location: the shelf will be in the hallway.

Russian distinguishes:

• в + accusative = motion into somewhere
• в + prepositional = location in/at somewhere

So:

• в коридор = into the hallway (movement)
• в коридоре = in the hallway (location)

In this sentence, the hallway is the place where the shelf will be hung, so Russian uses the prepositional case:

• в коридоре

Yes, Russian word order is flexible.

The sentence:

• Я принесла стремянку, чтобы повесить новую полку в коридоре.

is a neutral, natural order.

But other orders are possible, depending on emphasis:

• Стремянку я принесла, чтобы повесить новую полку в коридоре.
  Emphasizes that it was the stepladder that I brought.
• В коридоре я принесла стремянку, чтобы повесить новую полку.
  Sounds like the hallway is being highlighted, though this may suggest the bringing happened there.

Russian uses word order more for focus and emphasis than English does.

Grammatically, Russian often leaves this to context, but in this sentence the most natural reading is that в коридоре goes with новую полку / повесить:

• to hang a new shelf in the hallway

That is because semantically it makes the most sense: hallways commonly have shelves, and the location sounds like the place where the shelf will be installed.

If you wanted to make it clearer that the bringing happened in the hallway, you would usually rephrase the sentence.

Принесла is perfective and means brought as a completed action: the stepladder was successfully brought to the relevant place.

Несла is imperfective and would mean something more like was carrying or carried with focus on the process rather than the result.

Compare:

• Я принесла стремянку = I brought the stepladder
  Result matters.
• Я несла стремянку = I was carrying the stepladder
  Focus on the action in progress.

In this sentence, the completed result is important, so принесла is the natural choice.