Learn Hardcore Esperanto: Mia amikino ĉiam elektas novan sporton ĉiun jaron. - My female friend always chooses a new sport every year.

Whether that means girlfriend (romantic partner) or just female friend depends on context.
If you want to be very clear that it is a romantic partner, many speakers say koramikino (literally: heart‑friend‑female = girlfriend).

Why is it Mia amikino and not Mian amikinon?

Because Mia amikino is the subject of the sentence, and in normal Esperanto the subject does not take the -n ending.

Mia amikino = my (female) friend → subject → no -n
novan sporton = a new sport → direct object of elektas → takes -n

The -n ending is mainly for:

direct objects
movement to(ward) a place
some time expressions (like ĉiun jaron here)

Since Mia amikino is the one doing the choosing, it stays without -n.

Why is the verb elektas in the present tense if this happens every year?

In Esperanto, the present tense -as is also used for habitual actions, not only for things happening right now.

So:

Mia amikino ĉiam elektas novan sporton ĉiun jaron.
= My (female) friend always chooses a new sport every year. (habit, repeated action)

If you talked about the past, you would use:

elektis – chose / used to choose

About the future:

elektos – will choose

Where does ĉiam usually go? Could I move it to another place in the sentence?

Ĉiam normally comes before the verb, but word order is flexible in Esperanto.

All of these are grammatically correct, with small differences in emphasis:

Mia amikino ĉiam elektas novan sporton ĉiun jaron.
(neutral, very common)
Ĉiam mia amikino elektas novan sporton ĉiun jaron.
(slightly more emphasis on always she does this)
Mia amikino elektas ĉiam novan sporton ĉiun jaron.
(emphasis that what she chooses is always new)

Meaning stays basically the same; the choice is more about style and emphasis.

What is the purpose of the -n in novan sporton?

The -n marks the direct object of the verb.

elekti = to choose
What is being chosen? → novan sporton
So novan sporton must be in the accusative: novan sporton.

Because adjectives agree with the noun, both take -n:

adjective: nova → novan
noun: sporto → sporton

If it were the subject, it would be:

Nova sporto estas sana. – A new sport is healthy. (no -n)

Why is there no word for a in a new sport?

Esperanto has no indefinite article (a / an).

novan sporton can mean a new sport or one new sport, depending on context.

There is only one article: la = the.
You would use it only if you mean a specific sport that both speaker and listener know:

Mia amikino ĉiam elektas la novan sporton ĉiun jaron.
= My friend always chooses the new sport (that we already talked about) every year.

Why do both ĉiun and jaron have the -n ending in ĉiun jaron?

Here ĉiun jaron is a time expression: every year.

Esperanto often uses the accusative -n for time (when? how long? how often?):

ĉiun tagon – every day
la tutan semajnon – the whole week
pasintan jaron – last year

So ĉiun jaron literally means something like for each year / every year, and both ĉiu and jaro get -n because ĉiu is an adjective here and must agree with jaro:

ĉiu jaro – each year (as a subject)
ĉiun jaron – every year (as a time adverbial)

Could I say Ĉiun jaron mia amikino ĉiam elektas novan sporton instead? Is that still correct?

Yes, that is completely correct.

Esperanto word order is quite flexible. Moving ĉiun jaron to the beginning gives it a bit more emphasis:

Ĉiun jaron mia amikino ĉiam elektas novan sporton.
Every year, my (female) friend always chooses a new sport.

The grammar and basic meaning stay the same.

Is there a shorter or more compact way to say ĉiun jaron?

Yes, you can use the adverb ĉiujare:

Mia amikino ĉiam elektas novan sporton ĉiujare.
= My friend always chooses a new sport every year.

ĉiujare is built from:

ĉiu (each)
jaro (year)
adverb ending -e

Both ĉiun jaron and ĉiujare are correct and common; ĉiun jaron is a bit more transparent for beginners.

Can I say Mia amikino ĉiam elektas novajn sportojn in the plural?

Yes, but the meaning changes slightly:

novan sporton – one new sport
novajn sportojn – new sports (more than one)

So:

Mia amikino ĉiam elektas novan sporton ĉiun jaron.
→ She chooses one new sport each year.
Mia amikino ĉiam elektas novajn sportojn ĉiun jaron.
→ She chooses several new sports every year.

In the plural, both words change:

nova → novajn
sporto → sportojn

How would I say My female friend always chooses a new sport for me every year?

You can add por mi (for me):

Mia amikino ĉiam elektas novan sporton por mi ĉiun jaron.

Structure:

Mia amikino – my (female) friend (subject)
ĉiam elektas – always chooses
novan sporton – a new sport (direct object)
por mi – for me
ĉiun jaron – every year

You could also move por mi elsewhere:

Mia amikino por mi ĉiam elektas novan sporton ĉiun jaron.
Ĉiun jaron mia amikino ĉiam elektas por mi novan sporton.

All of these are correct; the differences are just in emphasis.