Kdyby nebyla řeka tak studená, moje dcera by tam plavala s námi.

It is a conditional sentence, specifically an unreal / hypothetical present condition.

It describes a situation that is not true now:

• the river is cold
• because of that, the daughter is not swimming with us

Czech expresses this with:

• kdyby in the if-clause
• by in the main clause

So the structure is roughly:

• Kdyby + past-form verb...
• ... by + past-form verb

This is the normal Czech way to say If X were..., Y would...

Because after kdyby, Czech normally uses the past-form of the verb, even when the meaning is about a present unreal situation.

So:

• Kdyby nebyla... = If it were not...
• not If it is not...

This is different from English, where learners may expect a present form after if.

Also, nebyla agrees with řeka, which is:

• singular
• feminine

That is why it is byla / nebyla, not byl / nebyl or bylo / nebylo.

Because both parts help mark the conditional.

• kdyby = if
• by plavala = would swim

Historically and grammatically, by is the conditional marker. In kdyby, it is built into the conjunction. In the main clause, it appears separately before the past-form verb.

So Czech often has:

• Kdyby...
• ... by ...

That is completely normal.

Because Czech forms the conditional with by plus the past-form participle.

So:

• plave = swims / is swimming
• by plavala = would swim

Again, plavala matches moje dcera, which is feminine singular.

Compare:

• Moje dcera plave. = My daughter swims / is swimming.
• Moje dcera by plavala. = My daughter would swim.

Because the sentence contains two feminine singular nouns, and Czech shows agreement clearly.

1. řeka is feminine singular
   So words connected with it are feminine singular:

   • nebyla
   • studená

2. dcera is also feminine singular
   So the verb referring to her is feminine singular:

   • plavala

This is a very important feature of Czech: verbs in the past/conditional and adjectives often agree with the noun’s gender and number.

Because the preposition s meaning with requires the instrumental case.

The pronoun my changes in the instrumental:

• my = we
• námi = us / with us after s

So:

• s námi = with us

This is something you simply have to learn with the preposition:

• s + instrumental

Examples:

• s kamarádem = with a friend
• s dcerou = with my daughter
• s námi = with us

Yes. That word order is very natural and many speakers might even prefer it.

Both are correct:

• Kdyby nebyla řeka tak studená...
• Kdyby řeka nebyla tak studená...

Czech word order is more flexible than English. Changing the order usually changes:

• emphasis
• rhythm
• style

The version with řeka earlier may sound a bit more neutral to many learners:

• Kdyby řeka nebyla tak studená, moje dcera by tam plavala s námi.

Tam means there.

Here it refers to the place of swimming, basically in that river / there in the water / there with us.

It is natural because Czech often uses location words like:

• tam = there
• tady / zde = here

It is not always strictly necessary, but it helps the sentence sound more complete and natural.

Depending on context, you might also hear something more explicit, such as:

• ... by v ní plavala s námi. = ... she would swim in it with us.

But tam is perfectly normal and idiomatic.

Tak means so in the sense of degree or intensity.

So:

• tak studená = so cold

It strengthens the adjective studená.

This is common in Czech:

• tak velký = so big
• tak malá = so small
• tak drahé = so expensive

In this sentence, tak helps explain the reason for the hypothetical situation: the river is so cold that the daughter is not swimming.

Not exactly.

• tak studená = so cold
• příliš studená = too cold

In many situations the practical meaning may be similar, but the nuance is different.

• tak studená just describes a high degree of coldness
• příliš studená explicitly means more cold than is acceptable / too cold

So this sentence does not literally say too cold, even if English might sometimes express the idea that way.

No, not in standard Czech if you want a true conditional meaning here.

You need:

• moje dcera by tam plavala

Without by, you get:

• moje dcera tam plavala

That would mean something like:

• my daughter swam there or, depending on context,
• a past statement rather than a hypothetical one

So by is essential for the would swim meaning.