natu no yoru ha, umi no tikaku de hanabi ga agarimasu.

Yes. の links two nouns, and here it means something like of or shows that one noun describes another.

• 夏 = summer
• 夜 = night
• 夏の夜 = summer night / night in summer

In natural English, we usually say summer night, not summer’s night, but the relationship is similar.

Because the whole phrase 夏の夜 is being treated as the topic:

• 夏の夜は = As for summer nights, ...

The speaker is not just talking about summer in general, but specifically about summer nights. So は comes after the full topic phrase.

It is mainly marking the topic, not just time by itself.

夏の夜は、... sets the scene as:

• When it comes to summer nights...
• On summer nights...
• As for summer nights...

In many sentences like this, a time expression can also become the topic, so は gives both a topical and contextual feeling.

で marks the place where an action or event happens.

Here:

• 海の近くで = near the sea, as the location where the fireworks go up / happen

Compare:

• で = where something happens
• に = destination, existence, or point of arrival in many cases

Since fireworks are an event, で is the natural choice.

近い is an i-adjective meaning near.

But 近く here is being used more like a noun or location word meaning:

• the vicinity
• the area near ...
• nearby

So:

• 海の近く = the area near the sea / near the sea

You can think of it as:

• 海に近い場所 = a place close to the sea
  but shortened naturally to:
• 海の近く

が marks 花火 as the grammatical subject of the verb 上がります.

So the core part is:

• 花火が上がります = fireworks go up / fireworks are launched

In event-style sentences, が often marks the thing that appears, happens, or undergoes the action/state.

Yes, literally 上がる means to rise / to go up.

With fireworks, Japanese often describes them from the perspective of them going up into the sky, so:

• 花火が上がる = fireworks go up / fireworks are launched

This is a very natural Japanese expression. English usually says fireworks go off or there are fireworks, but Japanese often focuses on the upward motion.

Because 上がる and 上げる are different verbs:

• 上がる = intransitive, to go up
• 上げる = transitive, to raise something

Here, the fireworks themselves are the subject:

• 花火が上がります = the fireworks go up

If you used 上げます, you would need someone as the subject:

• 人が花火を上げます = A person sets off / launches fireworks

It sounds most naturally like a general or habitual statement:

• On summer nights, fireworks go up near the sea.
• In summer nights, there are fireworks near the sea.

The non-past 上がります can describe:

• habitual events
• regular occurrences
• general truths
• future events, depending on context

Without more context, this sounds like a general seasonal scene.

It can mean either, depending on context. Japanese usually does not mark singular vs. plural the way English does.

So:

• 花火が上がります could literally be a firework goes up
• but in natural English here, fireworks go up is more likely

Because fireworks are commonly thought of as a display, the plural sense is very natural.

They are:

• 夏 = なつ
• の = の
• 夜 = よる
• は = written は, pronounced わ as a particle
• 海 = うみ
• の = の
• 近く = ちかく
• で = で
• 花火 = はなび
• が = が
• 上がります = あがります

So the full reading is:

なつ の よる は、うみ の ちかく で はなび が あがります。

The comma is mainly for readability and a pause.

It separates the topic-setting phrase:

• 夏の夜は、

from the rest of the sentence:

• 海の近くで花火が上がります。

Japanese commas are often more flexible than English commas. The sentence would still be understandable without it, but the comma makes the structure clearer.

A helpful breakdown is:

• 夏の夜は = topic / time setting
• 海の近くで = place where the event happens
• 花火が = subject
• 上がります = verb

So the structure is roughly:

[As for summer nights], [near the sea], [fireworks] [go up].

That is a very common Japanese pattern:

• Topic + place + subject + verb