Под липой летом всегда прохладно, и бабушка любит сидеть там в тени.

Because after под meaning under, Russian uses different cases depending on whether there is location or motion.

• под + Instrumental = location: under
• под + Accusative = motion toward/to under

Here the meaning is static location, so липа becomes instrumental singular:

• липа → липой

Compare:

• сидеть под липой = to sit under the linden tree
• идти под липу = to go under the linden tree

Летом is the instrumental singular of лето, but here it functions as an adverbial time expression meaning in summer.

Russian often uses seasons this way:

• зимой = in winter
• весной = in spring
• летом = in summer
• осенью = in autumn

So this is a very common pattern, and you should learn летом as a standard time expression.

In the present tense, Russian normally omits the verb to be.

So where English says:

• It is cool

Russian simply says:

• прохладно

And with всегда:

• всегда прохладно = it is always cool

This is completely normal Russian grammar.

Here прохладно is best understood as a predicative adverb or state word. It describes the general environment or condition, not a noun.

So:

• прохладный = cool, fairly cool — an adjective, used with nouns
  
  • прохладный день = a cool day
• прохладно = it is cool / coolly — used for the situation or atmosphere

In this sentence, there is no noun being described, so Russian uses прохладно.

Because this phrase describes location, not movement.

With в:

• в + Prepositional = in, inside, at a place
• в + Accusative = into, motion toward a place

Here the grandmother is sitting in the shade, so it is location:

• в тени

Compare:

• сидеть в тени = to sit in the shade
• идти в тень = to go into the shade

The noun changes like this:

• тень → в тени

After любит meaning likes, Russian normally uses the infinitive of the second verb.

So:

• любит сидеть = likes to sit

This is similar to English likes to sit.

You cannot normally use a second fully conjugated verb here in the same way. So любит сидит would be ungrammatical.

Because сидеть here describes a habitual, repeated, or ongoing kind of action: what she likes to do in general.

• любит сидеть = she likes sitting / likes to sit

The imperfective is natural because the sentence is about a regular preference, not a single completed action.

Там means there, and it points back to the place already mentioned: under the linden tree.

So it helps connect the two parts:

• Под липой...
• бабушка любит сидеть там...

Without там, the sentence would still be grammatical:

• бабушка любит сидеть в тени

But там adds a natural sense of there, in that spot.

Yes, Russian word order is flexible. The version in the sentence is natural and neutral.

• сидеть там в тени = to sit there in the shade

Other orders are possible, but they change emphasis slightly:

• там сидеть в тени = stronger emphasis on there
• сидеть в тени там = possible in context, but less neutral

Russian often moves words around for focus, rhythm, or style, not because the basic grammar changes.

Because и here joins two clauses, each with its own grammatical center:

1. Под липой летом всегда прохладно
2. бабушка любит сидеть там в тени

In Russian, when и connects two independent clauses, a comma is usually needed.

A learner may want to know the main word stress:

• под ли́пой
• ле́том
• всегда́
• прохла́дно
• ба́бушка
• лю́бит
• сиде́ть
• там
• в тени́

Stress matters in Russian, so it is useful to learn words together with their stressed syllables.