Já mám rýmu, ale moje sestra má jen kašel.

That’s right: Czech very often drops subject pronouns because the verb ending already tells you who the subject is.

So Mám rýmu is usually enough for I have a cold/runny nose.

Here, já is included for emphasis or contrast. It helps set up the comparison:

• Já mám rýmu
• ale moje sestra má jen kašel

So it feels a bit like:

• I have a cold, but my sister only has a cough.

If there is no special emphasis, many speakers would simply say:

• Mám rýmu, ale moje sestra má jen kašel.

Because after mít (to have), the thing you have is normally in the accusative case.

The noun rýma is feminine, and its forms are:

• nominative: rýma
• accusative: rýmu

So:

• rýma = the basic dictionary form
• mám rýmu = I have a cold/runny nose

This is a very common pattern in Czech:

• mám sestru
• mám knihu
• mám rýmu

Because kašel also stands in the accusative, but this noun happens to have the same form in nominative and accusative singular.

So:

• nominative: kašel
• accusative: kašel

That is normal for many masculine inanimate nouns in Czech.

So in this sentence:

• má jen kašel = she has only a cough

Both rýmu and kašel are objects of mít; they just decline differently.

Jen means only or just.

In this sentence:

• má jen kašel

it means that the sister has only a cough, and not something worse or additional.

So jen limits what follows. Here it focuses on kašel.

A very similar word is jenom, which also means only/just:

• má jen kašel
• má jenom kašel

Both are natural. Jen is simply shorter.

Yes, all three are possible, but they feel a little different.

• moje sestra = my sister
• má sestra = also my sister, but often a bit more literary, formal, or emphatic
• sestra = sister, if it is already clear whose sister you mean from context

In everyday spoken Czech, moje sestra is very common and natural.

So:

• Moje sestra má jen kašel sounds neutral and normal.
• Má sestra má jen kašel is correct, but may sound a bit more marked.
• Sestra má jen kašel is also possible if the context already makes it clear.

Yes, that version is perfectly natural.

Czech word order is more flexible than English, because endings carry a lot of grammatical information. The choice of word order often changes emphasis rather than basic meaning.

These are both fine:

• Já mám rýmu, ale moje sestra má jen kašel.
• Mám rýmu, ale moje sestra má jen kašel.

The first one puts a little more focus on já.

Also, the position of jen matters. In má jen kašel, the focus is on only a cough. If you move jen, the emphasis can shift.

Because ale means but, and in Czech it is normally preceded by a comma when it joins two clauses.

So:

• Já mám rýmu, ale moje sestra má jen kašel.

This is standard Czech punctuation.

That comma is very similar to English punctuation before but in a sentence like:

• I have a cold, but my sister only has a cough.

In Czech, the acute accent usually shows vowel length, not stress.

So:

• já has a long á
• mám has a long á
• rýmu has a long ý

That means you hold the vowel a bit longer.

Important point: Czech word stress is usually on the first syllable of the word, regardless of vowel length.

So:

• JÁ mám
• MÁM
• RÝmu
• SEStra
• KAšel

The accent mark does not tell you where the stress goes; it tells you the vowel is long.

Very often, yes. Czech commonly uses mít (to have) with symptoms and some common health problems.

For example:

• mám rýmu = I have a cold/runny nose
• mám kašel = I have a cough
• mám horečku = I have a fever

So this sentence is very natural Czech.

Also, rýma is not exactly the most general word for every kind of cold. It is more specifically a runny/stuffy nose kind of cold symptom. But in everyday use, English speakers will often see it translated simply as a cold depending on context.