Раз дождь закончился, давай хотя бы немного погуляем у реки.

Here раз does not mean one time or once.

At the start of a sentence, раз can be a conjunction meaning something like:

• since
• now that
• if, given that

So Раз дождь закончился... means roughly Since / now that the rain has stopped...

This use is fairly common in spoken Russian and sounds conversational.

Because Раз дождь закончился is a subordinate clause, and it is followed by the main clause:

• Раз дождь закончился, — subordinate clause
• давай хотя бы немного погуляем у реки. — main clause

Russian uses commas very regularly to separate clauses, often more consistently than English does.

Because дождь is the subject of the verb закончился.

In Russian, the thing that comes to an end is the grammatical subject:

• дождь закончился = the rain ended / stopped

So дождь stays in the nominative case.

English often says the rain stopped, and Russian works similarly here.

The -ся here is part of the verb закончиться, which means to end / to come to an end.

Compare:

• закончить что-то = to finish something
  
  • Я закончил работу = I finished the work
• закончиться = to end / to be over
  
  • Дождь закончился = The rain ended

So in this sentence, -ся does not make it passive in the English sense. It is simply the normal form of the intransitive verb meaning to come to an end.

Because it agrees with дождь, which is:

• singular
• masculine

Past tense verbs in Russian agree with gender and number:

• masculine: закончился
• feminine: закончилась
• neuter: закончилось
• plural: закончились

So:

• дождь закончился
• буря закончилась
• лето закончилось
• дожди закончились

Давай here is used to make a suggestion, like:

• let’s
• come on, let’s

So давай ... погуляем means let’s go for a walk.

This is a very common conversational pattern in Russian:

• давай + 1st person plural verb
• давай + infinitive in some contexts

Examples:

• Давай поедем домой. = Let’s go home.
• Давай поговорим. = Let’s talk.

In your sentence, давай makes the tone friendly and natural.

Погуляем is the 1st person plural future form of the perfective verb погулять.

In this structure, Russian often uses a 1st person plural verb to mean let’s do X:

• погуляем = let’s take a walk / let’s go for a walk

Why погулять instead of гулять?

• гулять = to walk, to stroll, to be out walking, more process-oriented
• погулять = to go for a walk for a while, more like a complete outing

So давай ... погуляем sounds very natural for suggesting a walk.

In this verb, по- often gives the idea of doing something for a while or as a short, bounded activity.

So:

• гулять = to walk / to stroll
• погулять = to go for a walk, to stroll for some time

That fits well with немного in the sentence:

• хотя бы немного погуляем = let’s at least walk a little / go for at least a short walk

So по- helps make the action sound limited and casual rather than open-ended.

Хотя бы means at least.

It softens the suggestion and gives the idea:

• if nothing more, then at least this
• even just a little would be good

So давай хотя бы немного погуляем is like:

• Let’s at least walk a little
• Let’s at least go for a short walk

It suggests the speaker is lowering expectations a bit, perhaps because of the weather or circumstances.

They do related jobs, but they are not the same.

• хотя бы = at least
• немного = a little / for a short time

Together they mean something like:

• at least a little
• at least for a bit

So:

• хотя бы lowers the demand
• немного limits the amount or duration

This combination is very natural in Russian.

У means by / near / beside, and it takes the genitive case.

So:

• река = nominative
• у реки = genitive, meaning by the river / near the river

Compare:

• у реки = by the river, near the river
• к реке = toward the river
• на реке = on the river, or sometimes at the river depending on context

In this sentence, у реки means the walk will happen near the river.

Russian often omits subject pronouns when the verb ending already makes the subject clear.

Here погуляем means we will walk / let’s walk, so мы is understood automatically.

Adding мы is possible, but usually unnecessary unless you want contrast or emphasis.

So Russian naturally prefers:

• Давай погуляем.

rather than

• Давай мы погуляем.

Yes, Russian word order is flexible, but the original order is very natural:

• Раз дождь закончился, давай хотя бы немного погуляем у реки.

This order presents the logic clearly:

1. the reason/background: the rain has stopped
2. the suggestion: let’s go for a walk
3. the limitation/softening: at least a little
4. the place: by the river

You could move things around for emphasis, for example:

• Раз дождь закончился, давай погуляем хотя бы немного у реки.
• У реки давай хотя бы немного погуляем.

But the original sounds smooth and idiomatic.

It is informal / conversational.

Clues:

• раз in the sense of since / now that is conversational
• давай is an informal way to suggest doing something together
• the whole sentence has a relaxed spoken tone

In more formal or neutral writing, you might see something more like:

• Поскольку дождь закончился, давай хотя бы немного погуляем у реки.

But for everyday speech, the original sentence sounds very natural.