Haar handschrift is klein, maar gelukkig wel leesbaar.

No. Haar here means her, and it agrees with the owner, not with the noun handschrift.

• haar handschrift = her handwriting
• zijn handschrift = his handwriting

Even though handschrift is a het-word (het handschrift), that does not affect the choice of haar. Possessive words in Dutch match the person who owns something, just like in English.

In this sentence, handschrift means handwriting: someone’s way of writing by hand.

So haar handschrift means:

• her handwriting style
• the way she writes by hand

Depending on context, handschrift can sometimes also refer to a manuscript, but here it clearly means handwriting.

Yes. In Dutch, klein handschrift or haar handschrift is klein is a normal way to say that someone writes in small letters.

So klein here does not mean that the handwriting is unimportant; it means:

• the letters are small
• the writing is tiny or compact

This is very similar to English Her handwriting is small.

Leesbaar is an adjective meaning legible or readable.

It is built from:

• lezen = to read
• -baar = a suffix meaning something like -able

So:

• leesbaar = readable / legible
• onleesbaar = unreadable / illegible

In this sentence, leesbaar is the natural adjective to describe handwriting.

Because they are used after the verb is. This is called predicative use.

Compare:

• Haar handschrift is klein.
• Haar handschrift is leesbaar.

But before a noun, you usually do add -e:

• het kleine handschrift
• het leesbare handschrift

So:

• after zijn / is → usually no -e
• before a noun → often -e

This wel adds contrastive emphasis. It does not simply mean well.

The sentence says:

• klein = that is a possible problem
• maar gelukkig wel leesbaar = but fortunately it is still / at least readable

So wel gives the sense of:

• actually
• still
• at least
• nevertheless

Without wel, the sentence would still be correct:

• Haar handschrift is klein, maar gelukkig leesbaar.

But with wel, the contrast feels stronger and more natural:

• small, but it is readable after all

Here gelukkig means fortunately, not happily.

Dutch gelukkig can be used in two common ways:

1. fortunately

   • Gelukkig regent het niet.
   • Fortunately, it isn’t raining.

2. related to happy

   • Ik ben gelukkig.
   • I am happy.

In your sentence, it clearly means fortunately:

• maar gelukkig wel leesbaar = but fortunately still legible

Because after maar (but), Dutch keeps normal main-clause word order, and the speaker places the adverbs before the final adjective for emphasis.

Structure:

• Haar handschrift = subject
• is = verb
• klein = adjective complement
• maar = coordinating conjunction
• gelukkig = sentence adverb
• wel = contrastive emphasis
• leesbaar = adjective complement

So the second part is understood as:

• maar [het is] gelukkig wel leesbaar

Dutch often leaves out repeated material when it is obvious.

Yes, you can think of it as an ellipsis: something is left out because it is already understood.

Full version:

• Haar handschrift is klein, maar het is gelukkig wel leesbaar.

More compact version:

• Haar handschrift is klein, maar gelukkig wel leesbaar.

Dutch does this very naturally, especially when the subject and verb would just repeat information from the first clause.

Because maar connects two contrasting parts of the sentence, much like but in English.

So the punctuation is very similar to English:

• Her handwriting is small, but fortunately still legible.

The comma helps separate the two contrasting ideas:

• problem: klein
• reassuring contrast: gelukkig wel leesbaar

Het handschrift is klein is grammatically correct, but it usually refers to the handwriting in a more general or less personal way. In your sentence, haar handschrift is more natural because you are talking about one person’s handwriting.

Ze heeft klein handschrift is not the normal form. You would usually say:

• Ze heeft een klein handschrift.
• Haar handschrift is klein.

Both are good, but they are structured differently:

• Ze heeft een klein handschrift = She has small handwriting.
• Haar handschrift is klein = Her handwriting is small.

Not exactly.

• leesbaar = legible, readable
• duidelijk = clear

Handwriting can be:

• leesbaar if you can read it
• duidelijk if it is clear and easy to understand visually or logically

For handwriting, leesbaar is the more precise word.
Someone’s writing may be klein, but still leesbaar.
If it is messy or hard to make out, it may be onleesbaar.

Yes, very often. That is one reason wel is useful here.

The sentence suggests:

• small handwriting often causes reading problems
• in this case, fortunately, that problem does not go too far

So there is an implied contrast like:

• It’s small, but fortunately not unreadable.

That subtle contrast is exactly what wel helps express.