Kad se u proljeće otopli, djeca se opet igraju u parku.

Yes. Kad and kada both mean when.

In this sentence, kad is the more usual everyday form. Kada is also correct, but it can sound a bit more formal or emphatic.

So these are both possible:

• Kad se u proljeće otopli...
• Kada se u proljeće otopli...

Both mean the same thing here.

Because the verb is otopliti se, which means to get warm, to warm up, or sometimes to thaw.

Here, se is just part of the verb. It does not mean itself.

So:

• otopliti se = to warm up / get warm
• Kad se otopli = When it warms up / when it gets warm

Also, English uses a dummy subject it in weather-type expressions, but Croatian usually does not. So Croatian says:

• Kad se otopli
  not literally When it warms itself, but When it gets warm

Because igrati se means to play in the sense of children playing.

This is a very important distinction:

• igrati se = to play
• igrati without se often means to play a sport/game/instrumental role, or to act

For example:

• Djeca se igraju. = The children are playing.
• Igraju nogomet. = They are playing football.
• Igra Hamleta. = He/She plays Hamlet.

So in your sentence, se belongs to igraju se because the meaning is the children are playing.

Because se is a clitic, and clitics in Croatian usually go near the beginning of the clause.

That is why Croatian prefers:

• Kad se u proljeće otopli

rather than:

• Kad u proljeće se otopli

The second version sounds unnatural in standard Croatian.

A useful learner rule is: words like se, sam, si, je, su, ću, etc. usually appear in the second position area of the clause, very early on.

Because Kad se u proljeće otopli is a subordinate clause placed before the main clause.

Croatian normally separates that kind of clause with a comma:

• Kad se u proljeće otopli, djeca se opet igraju u parku.

This is similar to English:

• When it gets warm in spring, the children play in the park again.

If the kad-clause comes first, the comma is standard.

Here u proljeće means in spring and functions as a time expression.

This is the normal idiomatic way to say it in Croatian. It is commonly analyzed as u + accusative in time expressions such as:

• u ponedjeljak = on Monday
• u jesen = in autumn
• u proljeće = in spring

Because proljeće is a neuter noun, its nominative and accusative singular forms look the same, so you do not see a different ending.

So even though the form is proljeće, the important thing for a learner is simply:

• u proljeće = in spring

Because u parku describes a location: in the park.

With u, Croatian usually distinguishes between:

• u + accusative for movement into something
• u + locative for being in something

So:

• Idem u park. = I am going to the park.
• Igram se u parku. = I am playing in the park.

In your sentence, the children are already there playing, so Croatian uses u parku.

This is very normal in Croatian.

After kad meaning when, Croatian often uses the present tense, even when English may think in future terms.

In particular:

• otopli is a present form with future-like meaning in this kind of clause
• igraju se is present because the sentence describes what happens habitually or generally

So the sense is:

• When it warms up in spring, the children play in the park again.

If you were talking about one specific future situation, Croatian might use a future form in the main clause, for example:

• Kad se u proljeće otopli, djeca će se opet igrati u parku.

But your original sentence sounds more like a general repeated situation.

Yes, exactly.

Croatian aspect is important here:

• otopliti se is perfective
• igrati se is imperfective

Why?

• otopliti se describes a change that reaches a point: to warm up / to become warm
• igrati se describes an ongoing or repeated activity: to play

So in this sentence:

• kad se otopli = when the warming-up happens / once it gets warm
• djeca se igraju = the children play / are playing as an activity

This is a very typical aspect combination in Croatian.

In meaning, djeca means children, so it is plural in sense, and it takes plural agreement:

• Djeca se igraju. = The children are playing.

For learners, the important thing is: treat djeca as plural for verbs and most agreement.

It is a special noun historically, so some of its forms are unusual:

• nominative: djeca
• genitive: djece
• dative/locative: djeci

But in your sentence, the main point is simple:

• djeca = children
• therefore igraju, not igra

Opet means again.

So djeca se opet igraju u parku means that the children are playing in the park again, typically after a period when they were not.

A close synonym is ponovno:

• Djeca se opet igraju u parku.
• Djeca se ponovno igraju u parku.

Both are correct, but opet is very common and everyday.

Yes, Croatian word order is fairly flexible, but the original sentence is very natural and neutral:

• Kad se u proljeće otopli, djeca se opet igraju u parku.

You could move some parts for emphasis, for example:

• Djeca se opet igraju u parku kad se u proljeće otopli.

But the clitics, especially se, still need to stay in their normal early position within their clause.

So while Croatian allows movement for emphasis or style, the original version is probably the best one for a learner to copy.