Если на улице слякоть, лучше сразу разуваться у двери.

In Russian, the verb to be is usually omitted in the present tense.

So:

• На улице слякоть = It is slushy outside / There is slush outside
• Literally: On the street slush

This is completely normal Russian.
Compare:

• Он дома = He is at home
• Здесь тепло = It is warm here

In the past or future, forms of быть do appear:

• На улице была слякоть = It was slushy outside
• На улице будет слякоть = It will be slushy outside

Слякоть is a noun meaning slush, muddy wet snow, or generally wet, dirty weather underfoot, especially when streets are messy from melting snow or rain.

It is often used for the kind of weather where shoes get dirty and floors get wet.

Grammar note:

• слякоть is a feminine noun
• It belongs to the group of feminine nouns ending in -ь

So in this sentence, слякоть is the thing being described as present outside.

На улице is the normal Russian expression for outside / in the street / outdoors.

Russian uses different prepositions in ways that do not always match English. Here:

• на улице = outside / on the street
• в доме = inside the house
• на работе = at work
• на кухне = in the kitchen in some contexts, but usually в кухне is not standard; normally на кухне is fixed idiomatic usage

So на улице is just the standard phrase you should learn as a set expression.

Because Если на улице слякоть is a subordinate clause introduced by если (if).

Russian normally separates this kind of clause with a comma:

• Если идёт дождь, мы останемся дома.
• Если на улице холодно, надень куртку.
• Если на улице слякоть, лучше сразу разуваться у двери.

So the comma is required here.

Here лучше means it’s better or better to.

Russian often uses лучше + infinitive to give general advice in an impersonal way:

• Лучше подождать. = It’s better to wait.
• Лучше не спорить. = It’s better not to argue.
• Лучше сразу разуваться у двери. = It’s better to take your shoes off right away at the door.

There is no explicit you, but the meaning is understood as a general recommendation to anyone in that situation.

Because the structure лучше + infinitive is a very common way to express advice.

So:

• лучше разуваться = it’s better to take off your shoes
• literally: better to unshoe oneself

This is similar to English better to... or it’s better to...

Other examples:

• Лучше лечь спать пораньше. = It’s better to go to bed early.
• Лучше сразу спросить. = It’s better to ask right away.

This is an aspect question.

• разуваться = imperfective
• разуться = perfective

In a sentence of general advice or habitual behavior, Russian often prefers the imperfective infinitive:

• Если грязно, лучше разуваться у двери. = general rule / what one should do in such situations

If you used разуться, it would sound more like a single completed action in a specific instance:

• Лучше сразу разуться у двери. = better to take your shoes off right away this one time

So разуваться fits the idea of a general recommendation very naturally.

The verb is reflexive because it means to take off one’s own shoes.

• разуть = to take someone’s shoes off
• разуться / разуваться = to take off one’s own shoes

The -ся often shows that the action comes back to the subject.

Compare:

• одеть ребёнка = to dress a child
• одеться = to get dressed
• обуть ребёнка = to put shoes on a child
• обуться = to put shoes on oneself
• разуться = to take off one’s own shoes

So the reflexive form is exactly what you want here.

Yes, you could, but the nuance is a little different.

• разуваться specifically means to take off your shoes
• снимать обувь literally means to remove footwear

Разуваться is more natural and compact in everyday speech when talking about taking off shoes at the entrance.

Compare:

• Лучше сразу разуваться у двери.
  Very natural, idiomatic.

• Лучше сразу снимать обувь у двери.
  Also understandable, but a bit less idiomatic in this exact context.

У usually takes the genitive case, and here двери is genitive singular of дверь.

• nominative: дверь = door
• genitive singular: двери

So:

• у двери = by the door / at the door

This phrase suggests the place right near the entrance, before walking farther into the home.

Not exactly.

• у двери = by the door / at the door
  the most natural choice here
• около двери = near the door
  possible, but a little more neutral or descriptive
• в двери usually means something like in the doorway / in the door and would not fit this sentence well

So у двери is the idiomatic way to say right at the entrance area.

Сразу means right away / immediately.

So:

• лучше разуваться у двери = it’s better to take your shoes off at the door
• лучше сразу разуваться у двери = it’s better to take your shoes off immediately at the door

It emphasizes not walking farther inside first, which matches the idea of muddy or slushy weather.

Yes, but it is not stated directly.

Russian often gives advice in an impersonal way, especially with:

• нужно
• надо
• можно
• нельзя
• лучше
• an infinitive construction

So this sentence really means something like:

• If it’s slushy outside, it’s better for people / for you / for one to take off shoes right away at the door.

It is general advice, not addressed to one specific person only.

Yes. Russian word order is fairly flexible.

The original:

• Если на улице слякоть, лучше сразу разуваться у двери.

You could also say:

• Лучше сразу разуваться у двери, если на улице слякоть.

Both are correct. The difference is mostly about emphasis:

• starting with Если... foregrounds the condition
• starting with Лучше... foregrounds the advice

The original version sounds very natural because it first sets the situation, then gives the recommendation.

It is neutral everyday Russian.

Nothing in it is especially formal or especially slangy. It would sound natural in:

• conversation
• advice from family members
• practical instructions
• everyday writing

It has a general, sensible, household tone.