Без стремянки мне трудно дотянуться до верхней полки над дверью.

Because без always takes the genitive case.

• Dictionary form: стремянка
• Genitive singular: стремянки

So без стремянки means without a stepladder.

This is a very common pattern:

• без сахара = without sugar
• без машины = without a car
• без друга = without a friend

Стремянка means a stepladder or folding ladder—the kind you can stand on indoors.

It is more specific than лестница, which can mean:

• a ladder
• a staircase

So in this sentence, стремянка is the natural word because the speaker needs something to stand on in order to reach a high shelf.

Russian often uses an impersonal construction here:

• мне = to me / for me (dative)
• трудно = difficult, hard

So мне трудно literally means it is hard for me.

This is the normal Russian way to express personal difficulty with many states or feelings:

• мне холодно = I am cold
• мне интересно = I am interested / it is interesting to me
• мне трудно понять = it is hard for me to understand

So the sentence is built like:

Без стремянки мне трудно + infinitive

Here трудно is not an adjective describing a noun. It is a predicative word (sometimes called a category of state word), used in impersonal sentences.

Compare:

• трудная работа = difficult work
  
  • here трудная is an adjective
• мне трудно работать = it is hard for me to work
  
  • here трудно is a predicative word meaning it is difficult

So in your sentence, трудно is functioning as the core idea it is hard.

Дотянуться means to reach in the physical sense of stretching enough to get to something.

It comes from the idea of stretching:

• тянуть = to pull
• тянуться = to stretch oneself
• дотянуться = to stretch far enough to reach

So this sentence is specifically about physical reach, not about obtaining or getting something in a broader sense.

A good mental translation is:

to be able to stretch up and reach

It is reflexive because the verb comes from тянуться, which means to stretch oneself / to reach out.

The -ся ending often shows that the action relates back to the subject. Here, the person is stretching themself in order to reach something.

So дотянуться is the normal form for reach (by stretching).

Because дотянуться normally goes with до + genitive.

This pattern means to reach as far as something:

• дотянуться до ручки = reach the handle
• дотянуться до потолка = reach the ceiling
• дотянуться до верхней полки = reach the top shelf

So in your sentence:

• до = up to / as far as
• верхней полки = upper shelf, in the genitive case

Because they are both part of the phrase after до, and до requires the genitive.

The noun is:

• полка = shelf

After до, it becomes:

• полки = of the shelf / to the shelf

The adjective must agree with the noun:

• верхняя полка = upper shelf
• до верхней полки = to the upper shelf

So both words are:

• feminine
• singular
• genitive

Because над normally takes the instrumental case.

• дверь = door
• instrumental singular: дверью

So:

• над дверью = above the door

This phrase describes where the shelf is located: it is the upper shelf above the door.

More examples:

• над столом = above the table
• над кроватью = above the bed
• над окном = above the window

It describes the shelf.

So the structure is:

• до верхней полки над дверью
  = to the upper shelf above the door

In other words, над дверью tells you which shelf we mean.

It does not mean the person is reaching above the door as a separate adverbial idea; it is part of the noun phrase identifying the shelf.

That word order puts the condition up front:

Without a stepladder, it’s hard for me...

Russian word order is flexible, so you could also say:

• Мне трудно дотянуться до верхней полки над дверью без стремянки.

That version is also understandable, but starting with Без стремянки highlights the important circumstance first.

So the original sentence sounds natural and emphasizes: without a stepladder.

Sometimes yes, but the nuance changes a little.

• дотянуться до... = to physically stretch enough to reach
• достать до... = to be able to reach as far as

In this sentence, дотянуться strongly emphasizes the act of stretching upward with your body or arm.

So Без стремянки мне трудно дотянуться... sounds very natural when talking about physical reach.

If you said достать, it would also be understandable, but дотянуться is especially vivid and specific here.

Because the idea is managing to reach something as a complete action.

• дотянуться = perfective, to succeed in reaching
• дотягиваться = imperfective, to be reaching / to manage to reach repeatedly or habitually

After трудно, Russian often uses the perfective infinitive when talking about whether a person can successfully do something:

• трудно понять
• трудно открыть
• трудно дотянуться

If you used дотягиваться, it would sound more like a repeated or habitual situation, and it is less natural here.

So трудно дотянуться is the normal choice.