kutu ga doro de kitanai node, genkan no soto de nugimasita.

In 靴が泥で汚い, 靴 is the subject of the description dirty with mud, so が is natural.

• が = marks what has the property
• は = would make shoes the topic, as in As for the shoes, they're dirty with mud
• を would mark the object of 脱ぐ in the second clause, not the subject of 汚い

So in the first clause, 靴が is correct.

Here, で is not marking the place where something happens. It shows the thing causing or making the state: dirty with mud.

So 泥で汚い means something like:

• dirty because of mud
• muddy / dirty with mud

This use of で also appears in sentences like:

• 汗でぬれた = got wet with sweat
• ほこりで汚れた = got dirty with dust

In Japanese, subordinate clauses often use the plain form even when the main sentence is polite.

So this pattern is very common:

• 汚いので、ぬぎました
• plain form + ので
• polite form in the main clause

汚いですので is possible in some formal situations, but it sounds stiffer and is much less common in everyday speech.

ので means because / since. It gives a reason in a softer, more explanatory way than から.

Compare:

• 汚いので、ぬぎました = Because they were dirty, I took them off.
• 汚いから、ぬぎました = Because they were dirty, I took them off.

Both are correct, but ので often sounds a little more gentle, objective, or matter-of-fact.

Because Japanese often leaves out words that are already obvious from context.

The full sentence could be:

• 靴が泥で汚いので、玄関の外で靴をぬぎました。

But repeating 靴 sounds unnecessary. Once the sentence has already established that we are talking about the shoes, the object of ぬぎました is understood.

ぬぎました is the polite past form of 脱ぐ（ぬぐ）, which means to take off something you are wearing.

So:

• dictionary form: 脱ぐ
• polite non-past: 脱ぎます
• polite past: 脱ぎました

In normal writing, this is often written with kanji as 脱ぎました, but hiragana is also common in learner materials.

It means outside the entrance / outside the entryway.

Breakdown:

• 玄関 = entrance, entryway, genkan
• の = links the nouns, so 玄関の外 = the outside of the entrance
• で = marks the place where the action happens

So 玄関の外でぬぎました means took them off outside the entrance.

Japanese often omits the subject when it is clear from context.

In English, you would usually say I took them off, but in Japanese, once the situation is understood, saying I is often unnecessary. The listener naturally understands that the person speaking is the one who took off the shoes.

Yes, it could.

• 汚い = dirty
• 汚れている = is dirty / has become dirty

Both work here, but the nuance is slightly different:

• 汚い describes the condition more directly
• 汚れている focuses a bit more on the state resulting from getting dirty

So 靴が泥で汚れているので would also be natural.

Yes. In Japanese homes, the 玄関 is the entrance area where people normally take off their shoes.

So this sentence feels very natural culturally. Saying the shoes were removed outside the entrance suggests the speaker did not want to bring muddy shoes even into the genkan area.