Не толкай меня, пожалуйста: на лестнице скользко.

This is a very common Russian aspect question.

• толкай is the imperfective imperative, from толкать
• толкни is the perfective imperative, from толкнуть

With negative commands, Russian very often prefers the imperfective when the meaning is a general warning or prohibition:

• Не толкай меня = Don’t push me
• Не кури здесь = Don’t smoke here
• Не спеши = Don’t hurry

Here the speaker means please don’t do that at all, not just don’t give me one single push, so не толкай sounds natural.

Толкай is the informal singular imperative of толкать (to push).

So it is used when speaking to:

• one person
• someone you address with ты

Compare:

• Не толкай меня = speaking to one person informally
• Не толкайте меня = speaking politely to one person or to several people

So the sentence is talking to one person informally.

Because толкать takes a direct object, and direct objects are usually in the accusative case.

The pronoun я changes like this:

• я = I
• меня = me

So:

• толкать меня = to push me

You can think of it as answering the question:

• толкать кого? = push whom?
• меня

Пожалуйста means please, and it often acts like a parenthetical politeness word.

That means it can appear in different places:

• Пожалуйста, не толкай меня.
• Не толкай меня, пожалуйста.
• Не толкай, пожалуйста, меня. (possible, but less natural here)

In this sentence, it softens the command and makes it more polite.

The commas are used because пожалуйста is inserted parenthetically:

• Не толкай меня, пожалуйста...

In everyday writing, you will very often see commas used this way with пожалуйста.

Because на лестнице expresses location, not movement.

Russian uses:

• на + prepositional for being on / at a place
• на + accusative for movement onto / toward a place

So:

• на лестнице = on the stairs / on the staircase
• на лестницу = onto the stairs / onto the staircase

Here the meaning is it is slippery on the stairs, so we need the prepositional case:

• на лестнице

Because Russian and English divide this idea differently.

Russian often uses лестница in the singular to mean:

• a staircase
• a stairway
• often simply the stairs

So:

• на лестнице very naturally means on the stairs

English often uses the plural stairs, but Russian commonly uses singular лестница for the whole structure as one unit.

Скользко is not describing a noun directly here. It is an impersonal predicative word, meaning something like:

• it is slippery
• things are slippery there

Russian often uses this kind of structure for states and conditions:

• Здесь холодно = It’s cold here
• На улице темно = It’s dark outside
• На лестнице скользко = It’s slippery on the stairs

If you said:

• Лестница скользкая

that would mean The staircase is slippery, with лестница as the grammatical subject.

Both are possible, but they are slightly different in feel:

• На лестнице скользко = focuses on the situation/location
• Лестница скользкая = focuses on the staircase itself

The colon introduces the reason/explanation for the request.

So the structure is:

• Don’t push me, please: it’s slippery on the stairs.

The second part explains why the speaker says the first part.

In English, we might also use:

• because
• a dash
• a full stop

Russian often uses a colon when the second clause explains or justifies the first one.

Yes. Russian word order is flexible, and different orders can sound more neutral, more emphatic, or more conversational.

For example:

• Не толкай меня, пожалуйста.
• Пожалуйста, не толкай меня.
• Меня не толкай, пожалуйста.
  This puts extra emphasis on me.

In your sentence, Не толкай меня, пожалуйста is a very natural, neutral order.

You would use the plural/polite imperative:

• Не толкайте меня, пожалуйста: на лестнице скользко.

So:

• не толкай = informal singular
• не толкайте = polite singular or plural

This is exactly like many other Russian imperatives:

• читай / читайте
• смотри / смотрите
• подожди / подождите

Yes, from an English point of view, it may feel that way.

Russian often uses impersonal sentences with no subject where English requires it:

• Холодно = It’s cold
• Жарко = It’s hot
• Скользко = It’s slippery

So in на лестнице скользко, Russian does not need a separate word for it. The idea is understood automatically.