Moje dcera umí vařit.

Because moje has to agree with dcera.

In Czech, possessive words like my change form to match the noun they go with in gender, number, and case.

• dcera = daughter, and it is feminine singular
• so my daughter here is moje dcera

So the important idea is: the possessive agrees with the thing possessed, not with the speaker.

It is in the nominative case because it is the subject of the sentence.

The sentence is about what my daughter can do, so dcera is the subject:

• Moje dcera = the one doing the action
• umí vařit = knows how to cook / can cook

Because dcera is nominative, moje is also in the matching nominative form.

Umí is the 3rd person singular present form of umět.

Here, umět means:

• to know how to
• to be able to because of skill or learned ability

So umí vařit means:

• she can cook
• she knows how to cook

Both are good English translations, but umět usually suggests ability based on knowledge or skill, not just possibility.

Because after umět, Czech normally uses an infinitive.

The pattern is:

umět + infinitive

Examples:

• umím plavat = I can swim / I know how to swim
• umíš zpívat = you can sing
• umí vařit = he/she can cook

So vařit stays in the basic dictionary form, to cook.

This is a very common question.

• umět = to know how to, to have a skill
• moci = can, may, to be able to because circumstances allow it

So:

• Moje dcera umí vařit = My daughter knows how to cook
• Moje dcera může vařit = My daughter can cook in the sense of is allowed to, has the opportunity to, or it is possible for her to cook

In this sentence, umět is the natural choice because cooking is treated as a skill.

It can mean both.

The Czech sentence Moje dcera umí vařit naturally covers both ideas in English:

• My daughter can cook
• My daughter knows how to cook

If you want the most precise explanation, umět is closer to know how to than to the broad English can.

Because Czech does not have articles like English a/an/the.

Czech usually shows definiteness through:

• context
• word order
• pronouns like moje
• the general situation

So dcera by itself does not need an article.
Here, moje already makes it clear whose daughter we mean.

You can, but the meaning becomes less specific.

• Moje dcera umí vařit = My daughter can cook
• Dcera umí vařit = The daughter / a daughter can cook, depending on context

Since Czech has no articles, dcera without moje needs context to tell the listener exactly which daughter is meant. If you want to clearly say my daughter, keep moje.

No, not normally.

Czech often leaves out subject pronouns because the verb ending already gives enough information.

• umí = he/she/it knows how / can

So Moje dcera umí vařit is completely natural without ona.

You would add ona only for emphasis or contrast, for example:

• Ona umí vařit, ale on ne. = She can cook, but he can’t.

Yes. Czech word order is more flexible than English word order.

The neutral order here is:

Moje dcera umí vařit.

But other orders are possible for emphasis, for example:

• Vařit moje dcera umí.
  This emphasizes cooking.

Even though the meaning stays similar, different word orders can sound more marked, contrastive, or stylistically specific. For a learner, Moje dcera umí vařit is the safest neutral version.

Because vařit is the more natural choice for a general ability.

• vařit is imperfective and refers to cooking in general
• uvařit is perfective and often suggests cooking something successfully to completion

So:

• umí vařit = she knows how to cook, in general
• umí uvařit polévku = she knows how to cook a soup / can make a soup successfully

In your sentence, the idea is a general skill, so vařit is the best fit.