Hur trött hon än är gillar hon ändå att fotografera familjen på födelsedagar.

It is a concessive pattern, used to mean no matter how ... or however ....

So:

• hur trött hon än är = no matter how tired she is
• literally, it is built from hur
  • adjective + subject + än
    • verb

This structure is very common in Swedish:

• Hur svårt det än är ... = No matter how difficult it is ...
• Vad han än säger ... = Whatever he says ...
• Var du än bor ... = Wherever you live ...

So here, the first part sets up a contrast: even in that situation, the rest of the sentence is still true.

Here, än does not mean the usual comparative than.

In this sentence, än is part of a fixed pattern with words like:

• hur = however / no matter how
• vad = whatever
• vem = whoever
• var = wherever

So in hur trött hon än är, än helps create the meaning no matter how tired she is.

This is different from comparative än, as in:

• Hon är tröttare än jag. = She is more tired than I am.

Same word, different function.

Ändå means still, nevertheless, or all the same.

You are right that hur trött hon än är already gives the idea of concession. Adding ändå in the main clause reinforces that contrast:

• Hur trött hon än är gillar hon att fotografera ...
• Hur trött hon än är gillar hon ändå att fotografera ...

Both are possible, but ändå makes the sentence sound more natural and more emphatic: even so, she still likes...

So the two parts work together:

• hur ... än ... = sets up no matter how / however
• ändå = marks the result as still / nevertheless

This is because Swedish is a V2 language in main clauses. V2 means the finite verb comes in the second position.

The sentence begins with the fronted clause:

• Hur trött hon än är

That whole chunk counts as position 1. Because of that, the finite verb of the main clause must come next:

• gillar

And then the subject comes after it:

• hon

So:

• Hur trött hon än är / gillar / hon ändå att fotografera ...

If the sentence started directly with the subject, then you would get normal order:

• Hon gillar ändå att fotografera familjen på födelsedagar.

Because that part is not a question. It may begin with hur, but it is not asking how tired is she? It is a clause meaning however tired she is.

So the normal order inside that clause is:

• hon = subject
• är = verb

That gives:

• hur trött hon än är

Compare:

• Hur trött är hon? = How tired is she?
  Here it is a real question, so verb-before-subject order appears.

• Hur trött hon än är ... = No matter how tired she is ...
  Here it is not a question, so subject-before-verb stays.

After gilla, Swedish often uses att + infinitive when talking about liking an activity.

So:

• gilla att fotografera = like to photograph / like photographing
• gilla att läsa = like to read
• gilla att laga mat = like cooking / like to cook

This is very natural Swedish.

English can often use either:

• likes to photograph
• likes photographing

Swedish usually uses att + infinitive in this kind of sentence.

Familjen is the definite form: the family.

In Swedish, definiteness is often shown by adding an ending to the noun:

• familj = family
• familjen = the family

Using en familj would mean a family, which sounds much less likely here unless you mean any random family.

In context, familjen probably means the family that is already understood from the situation, often something like her family or the family members. Swedish often leaves that kind of relationship to context if it is clear.

På is often used for events or occasions, so:

• på födelsedagar = at birthdays / on birthdays

The plural födelsedagar is used because the sentence is talking about a general habit, not one specific birthday.

So it means something like:

• she likes photographing the family whenever there are birthdays
• she likes photographing the family at birthday celebrations in general

Compare:

• på födelsedagar = on birthdays, in general
• på födelsedagen = on the birthday / at the birthday, one specific occasion

Yes, you could say:

• Även om hon är trött gillar hon ändå att fotografera familjen på födelsedagar.

That means roughly the same thing: Even if/although she is tired, she still likes photographing the family on birthdays.

The difference is mainly one of nuance:

• även om hon är trött = even if / although she is tired
• hur trött hon än är = no matter how tired she is

So hur trött hon än är is a bit stronger. It emphasizes that the degree of tiredness does not matter.

Because Swedish generally uses fewer commas than English.

In English, many writers would naturally put a comma after an introductory clause:

• No matter how tired she is, she still likes...

In Swedish, that comma is often left out unless it helps clarity or style. So:

• Hur trött hon än är gillar hon ändå ...

looks normal.

You may sometimes see a comma in similar sentences, especially for rhythm or emphasis, but it is not required in standard Swedish here.

Literally it is on birthdays, but in real use it often refers to the birthday occasion or birthday celebration more generally.

So depending on context, an English translation might sound more natural as:

• on birthdays
• at birthdays
• at family birthday celebrations

Swedish often uses på with event nouns in a broad way like this.

Yes. Swedish often relies on context more than English does in this kind of sentence.

So familjen literally means the family, but in context it may naturally refer to:

• her family
• the family
• the family members

If the situation already makes it clear whose family is meant, Swedish does not always need an extra possessive word.

If you wanted to make it explicit, you could say:

• sin familj = her own family (if it refers back to the subject)
• hennes familj = her family

But familjen is perfectly normal if the reference is already understood.