Hun er følsom over for kritik.

Because følsom is a predicate adjective. In Danish, adjectives often come after at være just like in English:

• Hun er følsom. = She is sensitive.

So the structure is:

• Hun = subject
• er = verb
• følsom = adjective describing the subject

Here, følsom means sensitive in an emotional or psychological sense: criticism affects her easily.

It can also mean sensitive in other contexts, depending on what follows:

• følsom over for kritik = sensitive to criticism
• følsom hud = sensitive skin
• følsom for lys = sensitive to light

So the word is broad, but in this sentence it clearly refers to a person’s reaction to criticism.

Because følsom over for is a very common Danish combination meaning sensitive to.

English uses to:

• sensitive to criticism

Danish usually uses:

• følsom over for kritik

So this is something you should learn as a fixed pattern:

• at være følsom over for noget = to be sensitive to something

In this expression, you will very often see over for written as two words:

• følsom over for kritik

You may also come across overfor in Danish in other contexts, especially in spatial meanings like opposite or facing. For a learner, the safest choice in this sentence is to keep the spelling you see here:

• over for

Usually, with kritik, the most natural phrasing is følsom over for kritik.

Danish does use følsom for in some other contexts, for example:

• følsom for kulde = sensitive to cold
• følsom for lys = sensitive to light

But when talking about a person’s emotional reaction to criticism, over for is the more idiomatic choice.

Because kritik is being used as a general, uncountable noun here.

So Danish says:

• kritik = criticism in general

Compare:

• Hun er følsom over for kritik. = She is sensitive to criticism.
• Hun er følsom over for kritikken. = She is sensitive to the criticism.

Adding -en would make it specific: a particular piece of criticism or a specific situation.

Yes. Like many Danish adjectives, it changes depending on gender/number and sometimes definiteness.

Examples:

• en følsom person = a sensitive person
• et følsomt barn = a sensitive child
• følsomme mennesker = sensitive people

In your sentence, it stays as følsom because it is describing hun, and after er with a singular common-gender subject, the basic form is used.

Er describes a state or characteristic:

• Hun er følsom over for kritik. = She is sensitive to criticism.

If you used bliver, it would mean becomes:

• Hun bliver følsom over for kritik. = She becomes sensitive to criticism.

So er suggests this is how she is, while bliver would suggest a change.

Because følsom is an adjective, not a feeling verb.

Danish can say:

• Hun føler sig trist. = She feels sad.

But here the idea is not she feels herself sensitive. It is simply a description of her character or tendency:

• Hun er følsom over for kritik.

So er is the natural verb.

A simple learner-friendly guide would be:

• Hun ≈ hoon
• er ≈ a very light air / er sound
• følsom ≈ roughly FURL-som or FURL-som, but with the Danish ø sound
• over for ≈ OH-vər for
• kritik ≈ kree-TEEK

A few useful notes:

• kritik has the stress on the last syllable: kri-TIK
• følsom has the stress on the first syllable
• the Danish ø in følsom does not exist in standard English, so that word may take practice

It depends on context, but it is often slightly evaluative.

It can sound:

• neutral: simply describing someone’s personality
• sympathetic: she is easily hurt by criticism
• mildly negative: she does not handle criticism well

So the sentence itself is not rude, but the tone depends on who says it and why.