В лесу рядом с фермой иногда появляется лиса.

Because here в means in in the sense of location, not motion.

• в лес = into the forest (direction, motion)
• в лесу = in the forest (location)

After в when you mean location, Russian normally uses the prepositional case.
The noun лес has the location form в лесу. This -у ending is a common special form with some masculine nouns for places.

So in this sentence, в лесу is correct because the fox appears in the forest, not into the forest.

Because the expression рядом с always takes the instrumental case.

So:

• ферма = dictionary form
• с фермой = instrumental singular

The phrase works as a unit:

• рядом с домом = next to the house
• рядом с дорогой = next to the road
• рядом с фермой = next to the farm

Even though с often means with, in рядом с it is just part of the fixed expression next to / near.

Russian present tense is often used for habitual or repeated actions, just like English appears in a sentence such as A fox sometimes appears...

Here, иногда means sometimes, so the sentence describes something that happens from time to time, not one single completed event.

That is why появляется is natural here:

• иногда появляется = sometimes appears

If you wanted a one-time future event, you would use a different form, such as появится.

Because иногда suggests a repeated or habitual event.

In Russian:

• imperfective is often used for repeated, ongoing, or general actions
• perfective is often used for a single completed event

So:

• иногда появляется = sometimes appears / shows up from time to time
• появится would sound more like will appear once / will show up

Since this sentence is about what happens on some occasions in general, the imperfective появляется is the right choice.

In Russian, word order is much more flexible than in English. The subject does not always have to come first.

Here, putting лиса at the end makes it the new or important information. The sentence is structured a bit like:

In the forest near the farm, there sometimes appears a fox.

So the end position gives лиса extra focus, as if introducing it into the scene.

A more neutral order is also possible:

• Лиса иногда появляется в лесу рядом с фермой.

But that version feels a little different in emphasis. The original sentence sounds more descriptive and scene-setting.

You know because of the verb form and the case.

• лиса is in the nominative singular form, which is the normal subject form
• появляется is 3rd person singular, agreeing with лиса

The other nouns are clearly in other cases:

• в лесу = prepositional
• с фермой = instrumental

So even though лиса comes last, it is still the subject.

Usually no. Russian often omits есть in ordinary present-tense existential or descriptive sentences.

English often says:

• There is a fox...
• A fox appears...

Russian simply says:

• ...появляется лиса

So Russian does not need an extra word meaning there is here.

It can mean either, depending on context, because Russian has no articles.

So лиса by itself may be understood as:

• a fox
• the fox

In this particular sentence, because the fox is introduced as something that appears from time to time, English will usually translate it as a fox.

But Russian itself does not mark that difference with an article.

Yes, those are possible, but the nuance is slightly different.

• рядом с фермой = right next to / beside the farm
• около фермы = near the farm
• возле фермы = near / by the farm

Also notice the grammar changes:

• рядом с фермой takes instrumental
• около фермы and возле фермы take genitive

So all of these can work, but рядом с often suggests a closer, more immediate position.

Yes, it can move. Russian adverbs are fairly mobile, and moving them changes the emphasis more than the basic meaning.

Original:

• В лесу рядом с фермой иногда появляется лиса.

Other possible orders:

• Иногда в лесу рядом с фермой появляется лиса.
• Лиса иногда появляется в лесу рядом с фермой.

All are grammatical. The original order first sets the scene (in the forest near the farm), then gives the time frequency (sometimes), and finally introduces the main new element (a fox).

Yes, появляется includes the ending -ся, which historically comes from a reflexive element, but here it is just part of the normal verb появляться.

You should learn it as a whole verb:

• появляться = to appear, to show up
• появиться = to appear, to show up (perfective)

In this verb, -ся does not literally mean it appears itself in the way an English learner might expect. It is simply part of the standard verb form.

It is singular because the sentence presents the event as one fox appearing at a time or as a typical recurring situation.

Russian often uses the singular for this kind of general statement:

• Иногда появляется лиса = sometimes a fox appears

If you meant that foxes in general appear there, or more than one may appear, you could use the plural:

• ...иногда появляются лисы.

That would change the image slightly from a fox shows up from time to time to foxes sometimes appear.

Here, рядом с фермой naturally modifies the location в лесу.

So the idea is:

• in the forest
• which is next to the farm

Together, they build one larger location phrase:

• in the forest next to the farm

You could think of it as layering location:

1. the event happens in the forest
2. that forest is near the farm

Yes. In many contexts, появляться can be translated as:

• appear
• show up
• turn up

The best English choice depends on tone:

• appears is more neutral and slightly more formal
• shows up is more conversational
• turns up is also natural in conversational English

In this sentence, appears is usually the safest choice, but shows up works well too if the tone is informal.